Definition df-itg1 25573

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 25568	. 2 class ∫1
2		vf	. . 3 setvar 𝑓
3		cr 11128	. . . . . 6 class ℝ
4		vg	. . . . . . 7 setvar 𝑔
5	4	cv 1539	. . . . . 6 class 𝑔
6	3, 3, 5	wf 6527	. . . . 5 wff 𝑔:ℝ⟶ℝ
7	5	crn 5655	. . . . . 6 class ran 𝑔
8		cfn 8959	. . . . . 6 class Fin
9	7, 8	wcel 2108	. . . . 5 wff ran 𝑔 ∈ Fin
10	5	ccnv 5653	. . . . . . . 8 class ◡𝑔
11		cc0 11129	. . . . . . . . . 10 class 0
12	11	csn 4601	. . . . . . . . 9 class {0}
13	3, 12	cdif 3923	. . . . . . . 8 class (ℝ ∖ {0})
14	10, 13	cima 5657	. . . . . . 7 class (◡𝑔 “ (ℝ ∖ {0}))
15		cvol 25416	. . . . . . 7 class vol
16	14, 15	cfv 6531	. . . . . 6 class (vol‘(◡𝑔 “ (ℝ ∖ {0})))
17	16, 3	wcel 2108	. . . . 5 wff (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ
18	6, 9, 17	w3a 1086	. . . 4 wff (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)
19		cmbf 25567	. . . 4 class MblFn
20	18, 4, 19	crab 3415	. . 3 class {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)}
21	2	cv 1539	. . . . . 6 class 𝑓
22	21	crn 5655	. . . . 5 class ran 𝑓
23	22, 12	cdif 3923	. . . 4 class (ran 𝑓 ∖ {0})
24		vx	. . . . . 6 setvar 𝑥
25	24	cv 1539	. . . . 5 class 𝑥
26	21	ccnv 5653	. . . . . . 7 class ◡𝑓
27	25	csn 4601	. . . . . . 7 class {𝑥}
28	26, 27	cima 5657	. . . . . 6 class (◡𝑓 “ {𝑥})
29	28, 15	cfv 6531	. . . . 5 class (vol‘(◡𝑓 “ {𝑥}))
30		cmul 11134	. . . . 5 class ·
31	25, 29, 30	co 7405	. . . 4 class (𝑥 · (vol‘(◡𝑓 “ {𝑥})))
32	23, 31, 24	csu 15702	. . 3 class Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(◡𝑓 “ {𝑥})))
33	2, 20, 32	cmpt 5201	. 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 1540	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  25627  itg1val  25636