Definition df-itg1 25848

Description: Define the Lebesgue integral for simple functions. A simple function is a finite linear combination of indicator functions for finitely measurable sets, whose assigned value is the sum of the measures of the sets times their respective weights. (Contributed by Mario Carneiro, 18-Jun-2014.)

Assertion

Ref	Expression
df-itg1	⊢ ∫1 = (𝑓 ∈ {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)} ↦ Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(◡𝑓 “ {𝑥}))))

Distinct variable group:   𝑓,𝑔,𝑥

Detailed syntax breakdown of Definition df-itg1

Step	Hyp	Ref	Expression
1		citg1 25843	. 2 class ∫1
2		vf	. . 3 setvar 𝑓
3		cr 11123	. . . . . 6 class ℝ
4		vg	. . . . . . 7 setvar 𝑔
5	4	cv 1569	. . . . . 6 class 𝑔
6	3, 3, 5	wf 6529	. . . . 5 wff 𝑔:ℝ⟶ℝ
7	5	crn 5656	. . . . . 6 class ran 𝑔
8		cfn 8952	. . . . . 6 class Fin
9	7, 8	wcel 2145	. . . . 5 wff ran 𝑔 ∈ Fin
10	5	ccnv 5654	. . . . . . . 8 class ◡𝑔
11		cc0 11124	. . . . . . . . . 10 class 0
12	11	csn 4584	. . . . . . . . 9 class {0}
13	3, 12	cdif 3896	. . . . . . . 8 class (ℝ ∖ {0})
14	10, 13	cima 5658	. . . . . . 7 class (◡𝑔 “ (ℝ ∖ {0}))
15		cvol 25691	. . . . . . 7 class vol
16	14, 15	cfv 6533	. . . . . 6 class (vol‘(◡𝑔 “ (ℝ ∖ {0})))
17	16, 3	wcel 2145	. . . . 5 wff (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ
18	6, 9, 17	w3a 1103	. . . 4 wff (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)
19		cmbf 25842	. . . 4 class MblFn
20	18, 4, 19	crab 3412	. . 3 class {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)}
21	2	cv 1569	. . . . . 6 class 𝑓
22	21	crn 5656	. . . . 5 class ran 𝑓
23	22, 12	cdif 3896	. . . 4 class (ran 𝑓 ∖ {0})
24		vx	. . . . . 6 setvar 𝑥
25	24	cv 1569	. . . . 5 class 𝑥
26	21	ccnv 5654	. . . . . . 7 class ◡𝑓
27	25	csn 4584	. . . . . . 7 class {𝑥}
28	26, 27	cima 5658	. . . . . 6 class (◡𝑓 “ {𝑥})
29	28, 15	cfv 6533	. . . . 5 class (vol‘(◡𝑓 “ {𝑥}))
30		cmul 11129	. . . . 5 class ·
31	25, 29, 30	co 7413	. . . 4 class (𝑥 · (vol‘(◡𝑓 “ {𝑥})))
32	23, 31, 24	csu 15773	. . 3 class Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(◡𝑓 “ {𝑥})))
33	2, 20, 32	cmpt 5186	. 2 class (𝑓 ∈ {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)} ↦ Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(◡𝑓 “ {𝑥}))))
34	1, 33	wceq 1570	1 wff ∫1 = (𝑓 ∈ {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)} ↦ Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(◡𝑓 “ {𝑥}))))

This definition is used by:  isi1f  25902  itg1val  25911