Definition df-itg1 25587

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 25582	. 2 class ∫1
2		vf	. . 3 setvar 𝑓
3		cr 11037	. . . . . 6 class ℝ
4		vg	. . . . . . 7 setvar 𝑔
5	4	cv 1541	. . . . . 6 class 𝑔
6	3, 3, 5	wf 6494	. . . . 5 wff 𝑔:ℝ⟶ℝ
7	5	crn 5632	. . . . . 6 class ran 𝑔
8		cfn 8893	. . . . . 6 class Fin
9	7, 8	wcel 2114	. . . . 5 wff ran 𝑔 ∈ Fin
10	5	ccnv 5630	. . . . . . . 8 class ◡𝑔
11		cc0 11038	. . . . . . . . . 10 class 0
12	11	csn 4567	. . . . . . . . 9 class {0}
13	3, 12	cdif 3886	. . . . . . . 8 class (ℝ ∖ {0})
14	10, 13	cima 5634	. . . . . . 7 class (◡𝑔 “ (ℝ ∖ {0}))
15		cvol 25430	. . . . . . 7 class vol
16	14, 15	cfv 6498	. . . . . 6 class (vol‘(◡𝑔 “ (ℝ ∖ {0})))
17	16, 3	wcel 2114	. . . . 5 wff (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ
18	6, 9, 17	w3a 1087	. . . 4 wff (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)
19		cmbf 25581	. . . 4 class MblFn
20	18, 4, 19	crab 3389	. . 3 class {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)}
21	2	cv 1541	. . . . . 6 class 𝑓
22	21	crn 5632	. . . . 5 class ran 𝑓
23	22, 12	cdif 3886	. . . 4 class (ran 𝑓 ∖ {0})
24		vx	. . . . . 6 setvar 𝑥
25	24	cv 1541	. . . . 5 class 𝑥
26	21	ccnv 5630	. . . . . . 7 class ◡𝑓
27	25	csn 4567	. . . . . . 7 class {𝑥}
28	26, 27	cima 5634	. . . . . 6 class (◡𝑓 “ {𝑥})
29	28, 15	cfv 6498	. . . . 5 class (vol‘(◡𝑓 “ {𝑥}))
30		cmul 11043	. . . . 5 class ·
31	25, 29, 30	co 7367	. . . 4 class (𝑥 · (vol‘(◡𝑓 “ {𝑥})))
32	23, 31, 24	csu 15648	. . 3 class Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(◡𝑓 “ {𝑥})))
33	2, 20, 32	cmpt 5166	. 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 1542	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 referenced by:  isi1f  25641  itg1val  25650