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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > measfrge0 | Structured version Visualization version GIF version |
Description: A measure is a function over its base to the positive extended reals. (Contributed by Thierry Arnoux, 26-Dec-2016.) |
Ref | Expression |
---|---|
measfrge0 | β’ (π β (measuresβπ) β π:πβΆ(0[,]+β)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | measbase 32860 | . . . 4 β’ (π β (measuresβπ) β π β βͺ ran sigAlgebra) | |
2 | ismeas 32862 | . . . 4 β’ (π β βͺ ran sigAlgebra β (π β (measuresβπ) β (π:πβΆ(0[,]+β) β§ (πββ ) = 0 β§ βπ¦ β π« π((π¦ βΌ Ο β§ Disj π₯ β π¦ π₯) β (πββͺ π¦) = Ξ£*π₯ β π¦(πβπ₯))))) | |
3 | 1, 2 | syl 17 | . . 3 β’ (π β (measuresβπ) β (π β (measuresβπ) β (π:πβΆ(0[,]+β) β§ (πββ ) = 0 β§ βπ¦ β π« π((π¦ βΌ Ο β§ Disj π₯ β π¦ π₯) β (πββͺ π¦) = Ξ£*π₯ β π¦(πβπ₯))))) |
4 | 3 | ibi 267 | . 2 β’ (π β (measuresβπ) β (π:πβΆ(0[,]+β) β§ (πββ ) = 0 β§ βπ¦ β π« π((π¦ βΌ Ο β§ Disj π₯ β π¦ π₯) β (πββͺ π¦) = Ξ£*π₯ β π¦(πβπ₯)))) |
5 | 4 | simp1d 1143 | 1 β’ (π β (measuresβπ) β π:πβΆ(0[,]+β)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3061 β c0 4286 π« cpw 4564 βͺ cuni 4869 Disj wdisj 5074 class class class wbr 5109 ran crn 5638 βΆwf 6496 βcfv 6500 (class class class)co 7361 Οcom 7806 βΌ cdom 8887 0cc0 11059 +βcpnf 11194 [,]cicc 13276 Ξ£*cesum 32690 sigAlgebracsiga 32771 measurescmeas 32858 |
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 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
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 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-ov 7364 df-esum 32691 df-meas 32859 |
This theorem is referenced by: measfn 32867 measvxrge0 32868 meascnbl 32882 measres 32885 measdivcst 32887 measdivcstALTV 32888 |
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