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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | mbflimsup 25701* | The limit supremum of a sequence of measurable real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵))) & ⊢ 𝐻 = (𝑚 ∈ ℝ ↦ sup((((𝑛 ∈ 𝑍 ↦ 𝐵) “ (𝑚[,)+∞)) ∩ ℝ*), ℝ*, < )) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) & ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐺 ∈ MblFn) | ||
| Theorem | mbflimlem 25702* | The pointwise limit of a sequence of measurable real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐶) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) & ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn) | ||
| Theorem | mbflim 25703* | The pointwise limit of a sequence of measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐶) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) & ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn) | ||
| Syntax | c0p 25704 | Extend class notation to include the zero polynomial. |
| class 0𝑝 | ||
| Definition | df-0p 25705 | Define the zero polynomial. (Contributed by Mario Carneiro, 19-Jun-2014.) |
| ⊢ 0𝑝 = (ℂ × {0}) | ||
| Theorem | 0pval 25706 | The zero function evaluates to zero at every point. (Contributed by Mario Carneiro, 23-Jul-2014.) |
| ⊢ (𝐴 ∈ ℂ → (0𝑝‘𝐴) = 0) | ||
| Theorem | 0plef 25707 | Two ways to say that the function 𝐹 on the reals is nonnegative. (Contributed by Mario Carneiro, 17-Aug-2014.) |
| ⊢ (𝐹:ℝ⟶(0[,)+∞) ↔ (𝐹:ℝ⟶ℝ ∧ 0𝑝 ∘r ≤ 𝐹)) | ||
| Theorem | 0pledm 25708 | Adjust the domain of the left argument to match the right, which works better in our theorems. (Contributed by Mario Carneiro, 28-Jul-2014.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℂ) & ⊢ (𝜑 → 𝐹 Fn 𝐴) ⇒ ⊢ (𝜑 → (0𝑝 ∘r ≤ 𝐹 ↔ (𝐴 × {0}) ∘r ≤ 𝐹)) | ||
| Theorem | isi1f 25709 | The predicate "𝐹 is a simple function". A simple function is a finite nonnegative linear combination of indicator functions for finitely measurable sets. We use the idiom 𝐹 ∈ dom ∫1 to represent this concept because ∫1 is the first preparation function for our final definition ∫ (see df-itg 25658); unlike that operator, which can integrate any function, this operator can only integrate simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.) |
| ⊢ (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))) | ||
| Theorem | i1fmbf 25710 | Simple functions are measurable. (Contributed by Mario Carneiro, 18-Jun-2014.) |
| ⊢ (𝐹 ∈ dom ∫1 → 𝐹 ∈ MblFn) | ||
| Theorem | i1ff 25711 | A simple function is a function on the reals. (Contributed by Mario Carneiro, 26-Jun-2014.) |
| ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ) | ||
| Theorem | i1frn 25712 | A simple function has finite range. (Contributed by Mario Carneiro, 26-Jun-2014.) |
| ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin) | ||
| Theorem | i1fima 25713 | Any preimage of a simple function is measurable. (Contributed by Mario Carneiro, 26-Jun-2014.) |
| ⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ 𝐴) ∈ dom vol) | ||
| Theorem | i1fima2 25714 | Any preimage of a simple function not containing zero has finite measure. (Contributed by Mario Carneiro, 26-Jun-2014.) |
| ⊢ ((𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴) → (vol‘(◡𝐹 “ 𝐴)) ∈ ℝ) | ||
| Theorem | i1fima2sn 25715 | Preimage of a singleton. (Contributed by Mario Carneiro, 26-Jun-2014.) |
| ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐴 ∈ (𝐵 ∖ {0})) → (vol‘(◡𝐹 “ {𝐴})) ∈ ℝ) | ||
| Theorem | i1fd 25716* | A simplified set of assumptions to show that a given function is simple. (Contributed by Mario Carneiro, 26-Jun-2014.) |
| ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → ran 𝐹 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑥}) ∈ dom vol) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑥})) ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐹 ∈ dom ∫1) | ||
| Theorem | i1f0rn 25717 | Any simple function takes the value zero on a set of unbounded measure, so in particular this set is not empty. (Contributed by Mario Carneiro, 18-Jun-2014.) |
| ⊢ (𝐹 ∈ dom ∫1 → 0 ∈ ran 𝐹) | ||
| Theorem | itg1val 25718* | The value of the integral on simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.) |
| ⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) = Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥})))) | ||
| Theorem | itg1val2 25719* | The value of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.) |
| ⊢ ((𝐹 ∈ dom ∫1 ∧ (𝐴 ∈ Fin ∧ (ran 𝐹 ∖ {0}) ⊆ 𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) → (∫1‘𝐹) = Σ𝑥 ∈ 𝐴 (𝑥 · (vol‘(◡𝐹 “ {𝑥})))) | ||
| Theorem | itg1cl 25720 | Closure of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.) |
| ⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) ∈ ℝ) | ||
| Theorem | itg1ge0 25721 | Closure of the integral on positive simple functions. (Contributed by Mario Carneiro, 19-Jun-2014.) |
| ⊢ ((𝐹 ∈ dom ∫1 ∧ 0𝑝 ∘r ≤ 𝐹) → 0 ≤ (∫1‘𝐹)) | ||
| Theorem | i1f0 25722 | The zero function is simple. (Contributed by Mario Carneiro, 18-Jun-2014.) |
| ⊢ (ℝ × {0}) ∈ dom ∫1 | ||
| Theorem | itg10 25723 | The zero function has zero integral. (Contributed by Mario Carneiro, 18-Jun-2014.) |
| ⊢ (∫1‘(ℝ × {0})) = 0 | ||
| Theorem | i1f1lem 25724* | Lemma for i1f1 25725 and itg11 25726. (Contributed by Mario Carneiro, 18-Jun-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 1, 0)) ⇒ ⊢ (𝐹:ℝ⟶{0, 1} ∧ (𝐴 ∈ dom vol → (◡𝐹 “ {1}) = 𝐴)) | ||
| Theorem | i1f1 25725* | Base case simple functions are indicator functions of measurable sets. (Contributed by Mario Carneiro, 18-Jun-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 1, 0)) ⇒ ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐹 ∈ dom ∫1) | ||
| Theorem | itg11 25726* | The integral of an indicator function is the volume of the set. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 1, 0)) ⇒ ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → (∫1‘𝐹) = (vol‘𝐴)) | ||
| Theorem | itg1addlem1 25727* | Decompose a preimage, which is always a disjoint union. (Contributed by Mario Carneiro, 25-Jun-2014.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
| ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ⊆ (◡𝐹 “ {𝑘})) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ dom vol) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (vol‘𝐵) ∈ ℝ) ⇒ ⊢ (𝜑 → (vol‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (vol‘𝐵)) | ||
| Theorem | i1faddlem 25728* | Decompose the preimage of a sum. (Contributed by Mario Carneiro, 19-Jun-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐺 ∈ dom ∫1) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (◡(𝐹 ∘f + 𝐺) “ {𝐴}) = ∪ 𝑦 ∈ ran 𝐺((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦}))) | ||
| Theorem | i1fmullem 25729* | Decompose the preimage of a product. (Contributed by Mario Carneiro, 19-Jun-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐺 ∈ dom ∫1) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (◡(𝐹 ∘f · 𝐺) “ {𝐴}) = ∪ 𝑦 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦}))) | ||
| Theorem | i1fadd 25730 | The sum of two simple functions is a simple function. (Contributed by Mario Carneiro, 18-Jun-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐺 ∈ dom ∫1) ⇒ ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ dom ∫1) | ||
| Theorem | i1fmul 25731 | The pointwise product of two simple functions is a simple function. (Contributed by Mario Carneiro, 5-Sep-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐺 ∈ dom ∫1) ⇒ ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ dom ∫1) | ||
| Theorem | itg1addlem2 25732* | Lemma for itg1add 25736. The function 𝐼 represents the pieces into which we will break up the domain of the sum. Since it is infinite only when both 𝑖 and 𝑗 are zero, we arbitrarily define it to be zero there to simplify the sums that are manipulated in itg1addlem4 25734 and itg1addlem5 25735. (Contributed by Mario Carneiro, 26-Jun-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐺 ∈ dom ∫1) & ⊢ 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))))) ⇒ ⊢ (𝜑 → 𝐼:(ℝ × ℝ)⟶ℝ) | ||
| Theorem | itg1addlem3 25733* | Lemma for itg1add 25736. (Contributed by Mario Carneiro, 26-Jun-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐺 ∈ dom ∫1) & ⊢ 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))))) ⇒ ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴𝐼𝐵) = (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵})))) | ||
| Theorem | itg1addlem4 25734* | Lemma for itg1add 25736. (Contributed by Mario Carneiro, 28-Jun-2014.) (Proof shortened by SN, 3-Oct-2024.) |
| ⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐺 ∈ dom ∫1) & ⊢ 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))))) & ⊢ 𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺)) ⇒ ⊢ (𝜑 → (∫1‘(𝐹 ∘f + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧))) | ||
| Theorem | itg1addlem5 25735* | Lemma for itg1add 25736. (Contributed by Mario Carneiro, 27-Jun-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐺 ∈ dom ∫1) & ⊢ 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))))) & ⊢ 𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺)) ⇒ ⊢ (𝜑 → (∫1‘(𝐹 ∘f + 𝐺)) = ((∫1‘𝐹) + (∫1‘𝐺))) | ||
| Theorem | itg1add 25736 | The integral of a sum of simple functions is the sum of the integrals. (Contributed by Mario Carneiro, 28-Jun-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐺 ∈ dom ∫1) ⇒ ⊢ (𝜑 → (∫1‘(𝐹 ∘f + 𝐺)) = ((∫1‘𝐹) + (∫1‘𝐺))) | ||
| Theorem | i1fmulclem 25737 | Decompose the preimage of a constant times a function. (Contributed by Mario Carneiro, 25-Jun-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (((𝜑 ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ) → (◡((ℝ × {𝐴}) ∘f · 𝐹) “ {𝐵}) = (◡𝐹 “ {(𝐵 / 𝐴)})) | ||
| Theorem | i1fmulc 25738 | A nonnegative constant times a simple function gives another simple function. (Contributed by Mario Carneiro, 25-Jun-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → ((ℝ × {𝐴}) ∘f · 𝐹) ∈ dom ∫1) | ||
| Theorem | itg1mulc 25739 | The integral of a constant times a simple function is the constant times the original integral. (Contributed by Mario Carneiro, 25-Jun-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (∫1‘((ℝ × {𝐴}) ∘f · 𝐹)) = (𝐴 · (∫1‘𝐹))) | ||
| Theorem | i1fres 25740* | The "restriction" of a simple function to a measurable subset is simple. (It's not actually a restriction because it is zero instead of undefined outside 𝐴.) (Contributed by Mario Carneiro, 29-Jun-2014.) |
| ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (𝐹‘𝑥), 0)) ⇒ ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐴 ∈ dom vol) → 𝐺 ∈ dom ∫1) | ||
| Theorem | i1fpos 25741* | The positive part of a simple function is simple. (Contributed by Mario Carneiro, 28-Jun-2014.) |
| ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ⇒ ⊢ (𝐹 ∈ dom ∫1 → 𝐺 ∈ dom ∫1) | ||
| Theorem | i1fposd 25742* | Deduction form of i1fposd 25742. (Contributed by Mario Carneiro, 6-Aug-2014.) |
| ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ 𝐴) ∈ dom ∫1) ⇒ ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(0 ≤ 𝐴, 𝐴, 0)) ∈ dom ∫1) | ||
| Theorem | i1fsub 25743 | The difference of two simple functions is a simple function. (Contributed by Mario Carneiro, 6-Aug-2014.) |
| ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → (𝐹 ∘f − 𝐺) ∈ dom ∫1) | ||
| Theorem | itg1sub 25744 | The integral of a difference of two simple functions. (Contributed by Mario Carneiro, 6-Aug-2014.) |
| ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → (∫1‘(𝐹 ∘f − 𝐺)) = ((∫1‘𝐹) − (∫1‘𝐺))) | ||
| Theorem | itg10a 25745* | The integral of a simple function supported on a nullset is zero. (Contributed by Mario Carneiro, 11-Aug-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → (vol*‘𝐴) = 0) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) = 0) ⇒ ⊢ (𝜑 → (∫1‘𝐹) = 0) | ||
| Theorem | itg1ge0a 25746* | The integral of an almost positive simple function is positive. (Contributed by Mario Carneiro, 11-Aug-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → (vol*‘𝐴) = 0) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → 0 ≤ (𝐹‘𝑥)) ⇒ ⊢ (𝜑 → 0 ≤ (∫1‘𝐹)) | ||
| Theorem | itg1lea 25747* | Approximate version of itg1le 25748. If 𝐹 ≤ 𝐺 for almost all 𝑥, then ∫1𝐹 ≤ ∫1𝐺. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → (vol*‘𝐴) = 0) & ⊢ (𝜑 → 𝐺 ∈ dom ∫1) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) ≤ (𝐺‘𝑥)) ⇒ ⊢ (𝜑 → (∫1‘𝐹) ≤ (∫1‘𝐺)) | ||
| Theorem | itg1le 25748 | If one simple function dominates another, then the integral of the larger is also larger. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.) |
| ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ∧ 𝐹 ∘r ≤ 𝐺) → (∫1‘𝐹) ≤ (∫1‘𝐺)) | ||
| Theorem | itg1climres 25749* | Restricting the simple function 𝐹 to the increasing sequence 𝐴(𝑛) of measurable sets whose union is ℝ yields a sequence of simple functions whose integrals approach the integral of 𝐹. (Contributed by Mario Carneiro, 15-Aug-2014.) |
| ⊢ (𝜑 → 𝐴:ℕ⟶dom vol) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ (𝐴‘(𝑛 + 1))) & ⊢ (𝜑 → ∪ ran 𝐴 = ℝ) & ⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0)) ⇒ ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫1‘𝐺)) ⇝ (∫1‘𝐹)) | ||
| Theorem | mbfi1fseqlem1 25750* | Lemma for mbfi1fseq 25756. (Contributed by Mario Carneiro, 16-Aug-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚))) ⇒ ⊢ (𝜑 → 𝐽:(ℕ × ℝ)⟶(0[,)+∞)) | ||
| Theorem | mbfi1fseqlem2 25751* | Lemma for mbfi1fseq 25756. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚))) & ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0))) ⇒ ⊢ (𝐴 ∈ ℕ → (𝐺‘𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0))) | ||
| Theorem | mbfi1fseqlem3 25752* | Lemma for mbfi1fseq 25756. (Contributed by Mario Carneiro, 16-Aug-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚))) & ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0))) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝐺‘𝐴):ℝ⟶ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))) | ||
| Theorem | mbfi1fseqlem4 25753* | Lemma for mbfi1fseq 25756. This lemma is not as interesting as it is long - it is simply checking that 𝐺 is in fact a sequence of simple functions, by verifying that its range is in (0...𝑛2↑𝑛) / (2↑𝑛) (which is to say, the numbers from 0 to 𝑛 in increments of 1 / (2↑𝑛)), and also that the preimage of each point 𝑘 is measurable, because it is equal to (-𝑛[,]𝑛) ∩ (◡𝐹 “ (𝑘[,)𝑘 + 1 / (2↑𝑛))) for 𝑘 < 𝑛 and (-𝑛[,]𝑛) ∩ (◡𝐹 “ (𝑘[,)+∞)) for 𝑘 = 𝑛. (Contributed by Mario Carneiro, 16-Aug-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚))) & ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0))) ⇒ ⊢ (𝜑 → 𝐺:ℕ⟶dom ∫1) | ||
| Theorem | mbfi1fseqlem5 25754* | Lemma for mbfi1fseq 25756. Verify that 𝐺 describes an increasing sequence of positive functions. (Contributed by Mario Carneiro, 16-Aug-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚))) & ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0))) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (0𝑝 ∘r ≤ (𝐺‘𝐴) ∧ (𝐺‘𝐴) ∘r ≤ (𝐺‘(𝐴 + 1)))) | ||
| Theorem | mbfi1fseqlem6 25755* | Lemma for mbfi1fseq 25756. Verify that 𝐺 converges pointwise to 𝐹, and wrap up the existential quantifier. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚))) & ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0))) ⇒ ⊢ (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))) | ||
| Theorem | mbfi1fseq 25756* | A characterization of measurability in terms of simple functions (this is an if and only if for nonnegative functions, although we don't prove it). Any nonnegative measurable function is the limit of an increasing sequence of nonnegative simple functions. This proof is an example of a poor de Bruijn factor - the formalized proof is much longer than an average hand proof, which usually just describes the function 𝐺 and "leaves the details as an exercise to the reader". (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) ⇒ ⊢ (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))) | ||
| Theorem | mbfi1flimlem 25757* | Lemma for mbfi1flim 25758. (Contributed by Mario Carneiro, 5-Sep-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) ⇒ ⊢ (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))) | ||
| Theorem | mbfi1flim 25758* | Any real measurable function has a sequence of simple functions that converges to it. (Contributed by Mario Carneiro, 5-Sep-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) ⇒ ⊢ (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ 𝐴 (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))) | ||
| Theorem | mbfmullem2 25759* | Lemma for mbfmul 25761. (Contributed by Mario Carneiro, 7-Sep-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐺 ∈ MblFn) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → 𝐺:𝐴⟶ℝ) & ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) & ⊢ (𝜑 → 𝑄:ℕ⟶dom ∫1) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ ℕ ↦ ((𝑄‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥)) ⇒ ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ MblFn) | ||
| Theorem | mbfmullem 25760 | Lemma for mbfmul 25761. (Contributed by Mario Carneiro, 7-Sep-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐺 ∈ MblFn) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → 𝐺:𝐴⟶ℝ) ⇒ ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ MblFn) | ||
| Theorem | mbfmul 25761 | The product of two measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐺 ∈ MblFn) ⇒ ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ MblFn) | ||
| Theorem | itg2lcl 25762* | The set of lower sums is a set of extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.) |
| ⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} ⇒ ⊢ 𝐿 ⊆ ℝ* | ||
| Theorem | itg2val 25763* | Value of the integral on nonnegative real functions. (Contributed by Mario Carneiro, 28-Jun-2014.) |
| ⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} ⇒ ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup(𝐿, ℝ*, < )) | ||
| Theorem | itg2l 25764* | Elementhood in the set 𝐿 of lower sums of the integral. (Contributed by Mario Carneiro, 28-Jun-2014.) |
| ⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} ⇒ ⊢ (𝐴 ∈ 𝐿 ↔ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔))) | ||
| Theorem | itg2lr 25765* | Sufficient condition for elementhood in the set 𝐿. (Contributed by Mario Carneiro, 28-Jun-2014.) |
| ⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} ⇒ ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝐺 ∘r ≤ 𝐹) → (∫1‘𝐺) ∈ 𝐿) | ||
| Theorem | xrge0f 25766 | A real function is a nonnegative extended real function if all its values are greater than or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 28-Jul-2014.) |
| ⊢ ((𝐹:ℝ⟶ℝ ∧ 0𝑝 ∘r ≤ 𝐹) → 𝐹:ℝ⟶(0[,]+∞)) | ||
| Theorem | itg2cl 25767 | The integral of a nonnegative real function is an extended real number. (Contributed by Mario Carneiro, 28-Jun-2014.) |
| ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) ∈ ℝ*) | ||
| Theorem | itg2ub 25768 | The integral of a nonnegative real function 𝐹 is an upper bound on the integrals of all simple functions 𝐺 dominated by 𝐹. (Contributed by Mario Carneiro, 28-Jun-2014.) |
| ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐺 ∈ dom ∫1 ∧ 𝐺 ∘r ≤ 𝐹) → (∫1‘𝐺) ≤ (∫2‘𝐹)) | ||
| Theorem | itg2leub 25769* | Any upper bound on the integrals of all simple functions 𝐺 dominated by 𝐹 is greater than (∫2‘𝐹), the least upper bound. (Contributed by Mario Carneiro, 28-Jun-2014.) |
| ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ*) → ((∫2‘𝐹) ≤ 𝐴 ↔ ∀𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 → (∫1‘𝑔) ≤ 𝐴))) | ||
| Theorem | itg2ge0 25770 | The integral of a nonnegative real function is greater than or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.) |
| ⊢ (𝐹:ℝ⟶(0[,]+∞) → 0 ≤ (∫2‘𝐹)) | ||
| Theorem | itg2itg1 25771 | The integral of a nonnegative simple function using ∫2 is the same as its value under ∫1. (Contributed by Mario Carneiro, 28-Jun-2014.) |
| ⊢ ((𝐹 ∈ dom ∫1 ∧ 0𝑝 ∘r ≤ 𝐹) → (∫2‘𝐹) = (∫1‘𝐹)) | ||
| Theorem | itg20 25772 | The integral of the zero function. (Contributed by Mario Carneiro, 28-Jun-2014.) |
| ⊢ (∫2‘(ℝ × {0})) = 0 | ||
| Theorem | itg2lecl 25773 | If an ∫2 integral is bounded above, then it is real. (Contributed by Mario Carneiro, 28-Jun-2014.) |
| ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ ∧ (∫2‘𝐹) ≤ 𝐴) → (∫2‘𝐹) ∈ ℝ) | ||
| Theorem | itg2le 25774 | If one function dominates another, then the integral of the larger is also larger. (Contributed by Mario Carneiro, 28-Jun-2014.) |
| ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐺:ℝ⟶(0[,]+∞) ∧ 𝐹 ∘r ≤ 𝐺) → (∫2‘𝐹) ≤ (∫2‘𝐺)) | ||
| Theorem | itg2const 25775* | Integral of a constant function. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
| ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ (0[,)+∞)) → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) = (𝐵 · (vol‘𝐴))) | ||
| Theorem | itg2const2 25776* | When the base set of a constant function has infinite volume, the integral is also infinite and vice-versa. (Contributed by Mario Carneiro, 30-Aug-2014.) |
| ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ+) → ((vol‘𝐴) ∈ ℝ ↔ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) ∈ ℝ)) | ||
| Theorem | itg2seq 25777* | Definitional property of the ∫2 integral: for any function 𝐹 there is a countable sequence 𝑔 of simple functions less than 𝐹 whose integrals converge to the integral of 𝐹. (This theorem is for the most part unnecessary in lieu of itg2i1fseq 25790, but unlike that theorem this one doesn't require 𝐹 to be measurable.) (Contributed by Mario Carneiro, 14-Aug-2014.) |
| ⊢ (𝐹:ℝ⟶(0[,]+∞) → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∘r ≤ 𝐹 ∧ (∫2‘𝐹) = sup(ran (𝑛 ∈ ℕ ↦ (∫1‘(𝑔‘𝑛))), ℝ*, < ))) | ||
| Theorem | itg2uba 25778* | Approximate version of itg2ub 25768. If 𝐹 approximately dominates 𝐺, then ∫1𝐺 ≤ ∫2𝐹. (Contributed by Mario Carneiro, 11-Aug-2014.) |
| ⊢ (𝜑 → 𝐹:ℝ⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐺 ∈ dom ∫1) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → (vol*‘𝐴) = 0) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐺‘𝑥) ≤ (𝐹‘𝑥)) ⇒ ⊢ (𝜑 → (∫1‘𝐺) ≤ (∫2‘𝐹)) | ||
| Theorem | itg2lea 25779* | Approximate version of itg2le 25774. If 𝐹 ≤ 𝐺 for almost all 𝑥, then ∫2𝐹 ≤ ∫2𝐺. (Contributed by Mario Carneiro, 11-Aug-2014.) |
| ⊢ (𝜑 → 𝐹:ℝ⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐺:ℝ⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → (vol*‘𝐴) = 0) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) ≤ (𝐺‘𝑥)) ⇒ ⊢ (𝜑 → (∫2‘𝐹) ≤ (∫2‘𝐺)) | ||
| Theorem | itg2eqa 25780* | Approximate equality of integrals. If 𝐹 = 𝐺 for almost all 𝑥, then ∫2𝐹 = ∫2𝐺. (Contributed by Mario Carneiro, 12-Aug-2014.) |
| ⊢ (𝜑 → 𝐹:ℝ⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐺:ℝ⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → (vol*‘𝐴) = 0) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) = (𝐺‘𝑥)) ⇒ ⊢ (𝜑 → (∫2‘𝐹) = (∫2‘𝐺)) | ||
| Theorem | itg2mulclem 25781 | Lemma for itg2mulc 25782. (Contributed by Mario Carneiro, 8-Jul-2014.) |
| ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → (∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) ≤ (𝐴 · (∫2‘𝐹))) | ||
| Theorem | itg2mulc 25782 | The integral of a nonnegative constant times a function is the constant times the integral of the original function. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
| ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ (0[,)+∞)) ⇒ ⊢ (𝜑 → (∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) = (𝐴 · (∫2‘𝐹))) | ||
| Theorem | itg2splitlem 25783* | Lemma for itg2split 25784. (Contributed by Mario Carneiro, 11-Aug-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ dom vol) & ⊢ (𝜑 → 𝐵 ∈ dom vol) & ⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) = 0) & ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ (0[,]+∞)) & ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐶, 0)) & ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐵, 𝐶, 0)) & ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝑈, 𝐶, 0)) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) & ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ) ⇒ ⊢ (𝜑 → (∫2‘𝐻) ≤ ((∫2‘𝐹) + (∫2‘𝐺))) | ||
| Theorem | itg2split 25784* | The ∫2 integral splits under an almost disjoint union. The proof avoids the use of itg2add 25794, which requires countable choice. (Contributed by Mario Carneiro, 11-Aug-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ dom vol) & ⊢ (𝜑 → 𝐵 ∈ dom vol) & ⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) = 0) & ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ (0[,]+∞)) & ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐶, 0)) & ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐵, 𝐶, 0)) & ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝑈, 𝐶, 0)) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) & ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ) ⇒ ⊢ (𝜑 → (∫2‘𝐻) = ((∫2‘𝐹) + (∫2‘𝐺))) | ||
| Theorem | itg2monolem1 25785* | Lemma for itg2mono 25788. We show that for any constant 𝑡 less than one, 𝑡 · ∫1𝐻 is less than 𝑆, and so ∫1𝐻 ≤ 𝑆, which is one half of the equality in itg2mono 25788. Consider the sequence 𝐴(𝑛) = {𝑥 ∣ 𝑡 · 𝐻 ≤ 𝐹(𝑛)}. This is an increasing sequence of measurable sets whose union is ℝ, and so 𝐻 ↾ 𝐴(𝑛) has an integral which equals ∫1𝐻 in the limit, by itg1climres 25749. Then by taking the limit in (𝑡 · 𝐻) ↾ 𝐴(𝑛) ≤ 𝐹(𝑛), we get 𝑡 · ∫1𝐻 ≤ 𝑆 as desired. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
| ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,)+∞)) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘r ≤ (𝐹‘(𝑛 + 1))) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) & ⊢ 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) & ⊢ (𝜑 → 𝑇 ∈ (0(,)1)) & ⊢ (𝜑 → 𝐻 ∈ dom ∫1) & ⊢ (𝜑 → 𝐻 ∘r ≤ 𝐺) & ⊢ (𝜑 → 𝑆 ∈ ℝ) & ⊢ 𝐴 = (𝑛 ∈ ℕ ↦ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻‘𝑥)) ≤ ((𝐹‘𝑛)‘𝑥)}) ⇒ ⊢ (𝜑 → (𝑇 · (∫1‘𝐻)) ≤ 𝑆) | ||
| Theorem | itg2monolem2 25786* | Lemma for itg2mono 25788. (Contributed by Mario Carneiro, 16-Aug-2014.) |
| ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,)+∞)) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘r ≤ (𝐹‘(𝑛 + 1))) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) & ⊢ 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) & ⊢ (𝜑 → 𝑃 ∈ dom ∫1) & ⊢ (𝜑 → 𝑃 ∘r ≤ 𝐺) & ⊢ (𝜑 → ¬ (∫1‘𝑃) ≤ 𝑆) ⇒ ⊢ (𝜑 → 𝑆 ∈ ℝ) | ||
| Theorem | itg2monolem3 25787* | Lemma for itg2mono 25788. (Contributed by Mario Carneiro, 16-Aug-2014.) |
| ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,)+∞)) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘r ≤ (𝐹‘(𝑛 + 1))) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) & ⊢ 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) & ⊢ (𝜑 → 𝑃 ∈ dom ∫1) & ⊢ (𝜑 → 𝑃 ∘r ≤ 𝐺) & ⊢ (𝜑 → ¬ (∫1‘𝑃) ≤ 𝑆) ⇒ ⊢ (𝜑 → (∫1‘𝑃) ≤ 𝑆) | ||
| Theorem | itg2mono 25788* | The Monotone Convergence Theorem for nonnegative functions. If {(𝐹‘𝑛):𝑛 ∈ ℕ} is a monotone increasing sequence of positive, measurable, real-valued functions, and 𝐺 is the pointwise limit of the sequence, then (∫2‘𝐺) is the limit of the sequence {(∫2‘(𝐹‘𝑛)):𝑛 ∈ ℕ}. (Contributed by Mario Carneiro, 16-Aug-2014.) |
| ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,)+∞)) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘r ≤ (𝐹‘(𝑛 + 1))) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) & ⊢ 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) ⇒ ⊢ (𝜑 → (∫2‘𝐺) = 𝑆) | ||
| Theorem | itg2i1fseqle 25789* | Subject to the conditions coming from mbfi1fseq 25756, the sequence of simple functions are all less than the target function 𝐹. (Contributed by Mario Carneiro, 17-Aug-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)))) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ⇒ ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → (𝑃‘𝑀) ∘r ≤ 𝐹) | ||
| Theorem | itg2i1fseq 25790* | Subject to the conditions coming from mbfi1fseq 25756, the integral of the sequence of simple functions converges to the integral of the target function. (Contributed by Mario Carneiro, 17-Aug-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)))) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃‘𝑚))) ⇒ ⊢ (𝜑 → (∫2‘𝐹) = sup(ran 𝑆, ℝ*, < )) | ||
| Theorem | itg2i1fseq2 25791* | In an extension to the results of itg2i1fseq 25790, if there is an upper bound on the integrals of the simple functions approaching 𝐹, then ∫2𝐹 is real and the standard limit relation applies. (Contributed by Mario Carneiro, 17-Aug-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)))) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃‘𝑚))) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∫1‘(𝑃‘𝑘)) ≤ 𝑀) ⇒ ⊢ (𝜑 → 𝑆 ⇝ (∫2‘𝐹)) | ||
| Theorem | itg2i1fseq3 25792* | Special case of itg2i1fseq2 25791: if the integral of 𝐹 is a real number, then the standard limit relation holds on the integrals of simple functions approaching 𝐹. (Contributed by Mario Carneiro, 17-Aug-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)))) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃‘𝑚))) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) ⇒ ⊢ (𝜑 → 𝑆 ⇝ (∫2‘𝐹)) | ||
| Theorem | itg2addlem 25793* | Lemma for itg2add 25794. (Contributed by Mario Carneiro, 17-Aug-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝐺 ∈ MblFn) & ⊢ (𝜑 → 𝐺:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ) & ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)))) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) & ⊢ (𝜑 → 𝑄:ℕ⟶dom ∫1) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑄‘𝑛) ∧ (𝑄‘𝑛) ∘r ≤ (𝑄‘(𝑛 + 1)))) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑄‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥)) ⇒ ⊢ (𝜑 → (∫2‘(𝐹 ∘f + 𝐺)) = ((∫2‘𝐹) + (∫2‘𝐺))) | ||
| Theorem | itg2add 25794 | The ∫2 integral is linear. (Measurability is an essential component of this theorem; otherwise consider the characteristic function of a nonmeasurable set and its complement.) (Contributed by Mario Carneiro, 17-Aug-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝐺 ∈ MblFn) & ⊢ (𝜑 → 𝐺:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ) ⇒ ⊢ (𝜑 → (∫2‘(𝐹 ∘f + 𝐺)) = ((∫2‘𝐹) + (∫2‘𝐺))) | ||
| Theorem | itg2gt0 25795* | If the function 𝐹 is strictly positive on a set of positive measure, then the integral of the function is positive. (Contributed by Mario Carneiro, 30-Aug-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ dom vol) & ⊢ (𝜑 → 0 < (vol‘𝐴)) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 < (𝐹‘𝑥)) ⇒ ⊢ (𝜑 → 0 < (∫2‘𝐹)) | ||
| Theorem | itg2cnlem1 25796* | Lemma for itgcn 25880. (Contributed by Mario Carneiro, 30-Aug-2014.) |
| ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) ⇒ ⊢ (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))), ℝ*, < ) = (∫2‘𝐹)) | ||
| Theorem | itg2cnlem2 25797* | Lemma for itgcn 25880. (Contributed by Mario Carneiro, 31-Aug-2014.) |
| ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → ¬ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑀, (𝐹‘𝑥), 0))) ≤ ((∫2‘𝐹) − (𝐶 / 2))) ⇒ ⊢ (𝜑 → ∃𝑑 ∈ ℝ+ ∀𝑢 ∈ dom vol((vol‘𝑢) < 𝑑 → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝑢, (𝐹‘𝑥), 0))) < 𝐶)) | ||
| Theorem | itg2cn 25798* | A sort of absolute continuity of the Lebesgue integral (this is the core of ftc1a 26078 which is about actual absolute continuity). (Contributed by Mario Carneiro, 1-Sep-2014.) |
| ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑑 ∈ ℝ+ ∀𝑢 ∈ dom vol((vol‘𝑢) < 𝑑 → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝑢, (𝐹‘𝑥), 0))) < 𝐶)) | ||
| Theorem | ibllem 25799 | Conditioned equality theorem for the if statement. (Contributed by Mario Carneiro, 31-Jul-2014.) |
| ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) | ||
| Theorem | isibl 25800* | The predicate "𝐹 is integrable". The "integrable" predicate corresponds roughly to the range of validity of ∫𝐴𝐵 d𝑥, which is to say that the expression ∫𝐴𝐵 d𝑥 doesn't make sense unless (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
| ⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘)))) & ⊢ (𝜑 → dom 𝐹 = 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ∈ 𝐿1 ↔ (𝐹 ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘𝐺) ∈ ℝ))) | ||
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