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Mirrors > Home > MPE Home > Th. List > Mathboxes > measbase | Structured version Visualization version GIF version |
Description: The base set of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
Ref | Expression |
---|---|
measbase | ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6944 | . 2 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ dom measures) | |
2 | vex 3482 | . . . . 5 ⊢ 𝑠 ∈ V | |
3 | ovex 7464 | . . . . 5 ⊢ (0[,]+∞) ∈ V | |
4 | mapex 7962 | . . . . 5 ⊢ ((𝑠 ∈ V ∧ (0[,]+∞) ∈ V) → {𝑚 ∣ 𝑚:𝑠⟶(0[,]+∞)} ∈ V) | |
5 | 2, 3, 4 | mp2an 692 | . . . 4 ⊢ {𝑚 ∣ 𝑚:𝑠⟶(0[,]+∞)} ∈ V |
6 | simp1 1135 | . . . . 5 ⊢ ((𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦))) → 𝑚:𝑠⟶(0[,]+∞)) | |
7 | 6 | ss2abi 4077 | . . . 4 ⊢ {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦)))} ⊆ {𝑚 ∣ 𝑚:𝑠⟶(0[,]+∞)} |
8 | 5, 7 | ssexi 5328 | . . 3 ⊢ {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦)))} ∈ V |
9 | df-meas 34177 | . . 3 ⊢ measures = (𝑠 ∈ ∪ ran sigAlgebra ↦ {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦)))}) | |
10 | 8, 9 | dmmpti 6713 | . 2 ⊢ dom measures = ∪ ran sigAlgebra |
11 | 1, 10 | eleqtrdi 2849 | 1 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 {cab 2712 ∀wral 3059 Vcvv 3478 ∅c0 4339 𝒫 cpw 4605 ∪ cuni 4912 Disj wdisj 5115 class class class wbr 5148 dom cdm 5689 ran crn 5690 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ωcom 7887 ≼ cdom 8982 0cc0 11153 +∞cpnf 11290 [,]cicc 13387 Σ*cesum 34008 sigAlgebracsiga 34089 measurescmeas 34176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-meas 34177 |
This theorem is referenced by: measfrge0 34184 measvnul 34187 measvun 34190 measxun2 34191 measun 34192 measvuni 34195 measssd 34196 measunl 34197 measiuns 34198 measiun 34199 meascnbl 34200 measinblem 34201 measinb 34202 measinb2 34204 measdivcst 34205 measdivcstALTV 34206 aean 34225 domprobsiga 34393 prob01 34395 probfinmeasb 34410 probfinmeasbALTV 34411 probmeasb 34412 |
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