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Definition df-sitm 30724
Description: Define the integral metric for simple functions, as the integral of the distances between the function values. Since distances take nonnegative values in *, the range structure for this integral is (ℝ*𝑠s (0[,]+∞)). See definition 2.3.1 of [Bogachev] p. 116. (Contributed by Thierry Arnoux, 22-Oct-2017.)
Assertion
Ref Expression
df-sitm sitm = (𝑤 ∈ V, 𝑚 ran measures ↦ (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓𝑓 (dist‘𝑤)𝑔))))
Distinct variable group:   𝑓,𝑔,𝑚,𝑤

Detailed syntax breakdown of Definition df-sitm
StepHypRef Expression
1 csitm 30721 . 2 class sitm
2 vw . . 3 setvar 𝑤
3 vm . . 3 setvar 𝑚
4 cvv 3398 . . 3 class V
5 cmeas 30589 . . . . 5 class measures
65crn 5319 . . . 4 class ran measures
76cuni 4637 . . 3 class ran measures
8 vf . . . 4 setvar 𝑓
9 vg . . . 4 setvar 𝑔
102cv 1636 . . . . . 6 class 𝑤
113cv 1636 . . . . . 6 class 𝑚
12 csitg 30722 . . . . . 6 class sitg
1310, 11, 12co 6877 . . . . 5 class (𝑤sitg𝑚)
1413cdm 5318 . . . 4 class dom (𝑤sitg𝑚)
158cv 1636 . . . . . 6 class 𝑓
169cv 1636 . . . . . 6 class 𝑔
17 cds 16165 . . . . . . . 8 class dist
1810, 17cfv 6104 . . . . . . 7 class (dist‘𝑤)
1918cof 7128 . . . . . 6 class 𝑓 (dist‘𝑤)
2015, 16, 19co 6877 . . . . 5 class (𝑓𝑓 (dist‘𝑤)𝑔)
21 cxrs 16368 . . . . . . 7 class *𝑠
22 cc0 10224 . . . . . . . 8 class 0
23 cpnf 10359 . . . . . . . 8 class +∞
24 cicc 12399 . . . . . . . 8 class [,]
2522, 23, 24co 6877 . . . . . . 7 class (0[,]+∞)
26 cress 16072 . . . . . . 7 class s
2721, 25, 26co 6877 . . . . . 6 class (ℝ*𝑠s (0[,]+∞))
2827, 11, 12co 6877 . . . . 5 class ((ℝ*𝑠s (0[,]+∞))sitg𝑚)
2920, 28cfv 6104 . . . 4 class (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓𝑓 (dist‘𝑤)𝑔))
308, 9, 14, 14, 29cmpt2 6879 . . 3 class (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓𝑓 (dist‘𝑤)𝑔)))
312, 3, 4, 7, 30cmpt2 6879 . 2 class (𝑤 ∈ V, 𝑚 ran measures ↦ (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓𝑓 (dist‘𝑤)𝑔))))
321, 31wceq 1637 1 wff sitm = (𝑤 ∈ V, 𝑚 ran measures ↦ (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ*𝑠s (0[,]+∞))sitg𝑚)‘(𝑓𝑓 (dist‘𝑤)𝑔))))
Colors of variables: wff setvar class
This definition is referenced by:  sitmval  30742
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