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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 6925 | . 2 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ dom measures) | |
2 | vex 3479 | . . . . 5 ⊢ 𝑠 ∈ V | |
3 | ovex 7437 | . . . . 5 ⊢ (0[,]+∞) ∈ V | |
4 | mapex 8822 | . . . . 5 ⊢ ((𝑠 ∈ V ∧ (0[,]+∞) ∈ V) → {𝑚 ∣ 𝑚:𝑠⟶(0[,]+∞)} ∈ V) | |
5 | 2, 3, 4 | mp2an 691 | . . . 4 ⊢ {𝑚 ∣ 𝑚:𝑠⟶(0[,]+∞)} ∈ V |
6 | simp1 1137 | . . . . 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 4062 | . . . 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 5321 | . . 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 33132 | . . 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 6691 | . 2 ⊢ dom measures = ∪ ran sigAlgebra |
11 | 1, 10 | eleqtrdi 2844 | 1 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 {cab 2710 ∀wral 3062 Vcvv 3475 ∅c0 4321 𝒫 cpw 4601 ∪ cuni 4907 Disj wdisj 5112 class class class wbr 5147 dom cdm 5675 ran crn 5676 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 ωcom 7850 ≼ cdom 8933 0cc0 11106 +∞cpnf 11241 [,]cicc 13323 Σ*cesum 32963 sigAlgebracsiga 33044 measurescmeas 33131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 df-ov 7407 df-meas 33132 |
This theorem is referenced by: measfrge0 33139 measvnul 33142 measvun 33145 measxun2 33146 measun 33147 measvuni 33150 measssd 33151 measunl 33152 measiuns 33153 measiun 33154 meascnbl 33155 measinblem 33156 measinb 33157 measinb2 33159 measdivcst 33160 measdivcstALTV 33161 aean 33180 domprobsiga 33348 prob01 33350 probfinmeasb 33365 probfinmeasbALTV 33366 probmeasb 33367 |
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