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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 6696 | . 2 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ dom measures) | |
2 | vex 3497 | . . . . 5 ⊢ 𝑠 ∈ V | |
3 | ovex 7183 | . . . . 5 ⊢ (0[,]+∞) ∈ V | |
4 | mapex 8406 | . . . . 5 ⊢ ((𝑠 ∈ V ∧ (0[,]+∞) ∈ V) → {𝑚 ∣ 𝑚:𝑠⟶(0[,]+∞)} ∈ V) | |
5 | 2, 3, 4 | mp2an 690 | . . . 4 ⊢ {𝑚 ∣ 𝑚:𝑠⟶(0[,]+∞)} ∈ V |
6 | simp1 1132 | . . . . 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 4042 | . . . 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 5218 | . . 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 31450 | . . 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 6486 | . 2 ⊢ dom measures = ∪ ran sigAlgebra |
11 | 1, 10 | eleqtrdi 2923 | 1 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {cab 2799 ∀wral 3138 Vcvv 3494 ∅c0 4290 𝒫 cpw 4538 ∪ cuni 4831 Disj wdisj 5023 class class class wbr 5058 dom cdm 5549 ran crn 5550 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ωcom 7574 ≼ cdom 8501 0cc0 10531 +∞cpnf 10666 [,]cicc 12735 Σ*cesum 31281 sigAlgebracsiga 31362 measurescmeas 31449 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7153 df-meas 31450 |
This theorem is referenced by: measfrge0 31457 measvnul 31460 measvun 31463 measxun2 31464 measun 31465 measvuni 31468 measssd 31469 measunl 31470 measiuns 31471 measiun 31472 meascnbl 31473 measinblem 31474 measinb 31475 measinb2 31477 measdivcst 31478 measdivcstALTV 31479 aean 31498 mbfmbfm 31511 domprobsiga 31664 prob01 31666 probfinmeasb 31681 probfinmeasbALTV 31682 probmeasb 31683 |
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