A map, [ilmath]f:(A,\mathcal{A})\rightarrow(F,\mathcal{F})[/ilmath], is [ilmath]\mathcal{A}/\mathcal{F} [/ilmath] measurable if and only if for some generator [ilmath]\mathcal{F}_0[/ilmath] of [ilmath]\mathcal{F} [/ilmath] we have [ilmath]\forall S\in\mathcal{F}_0[f^{-1}(S)\in\mathcal{A}][/ilmath]

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A map from a [ilmath]\sigma[/ilmath]-algebra [ilmath](A,\mathcal{A})[/ilmath] to another [ilmath]\sigma[/ilmath]-algebra [ilmath](F,\mathcal{F})[/ilmath], [ilmath]f:A\rightarrow F[/ilmath], is [ilmath]\mathcal{A}/\mathcal{F} [/ilmath] measurable if and only if for some generator, [ilmath]\mathcal{G} [/ilmath], of [ilmath]\mathcal{F} [/ilmath][Note 1] we have[1][2]:

  • [ilmath]\forall S\in\mathcal{G}[f^{-1}(S)\in\mathcal{A}][/ilmath]

Which we may alternatively write (for brevity, see: abuses of the implies-subset relation) as:

  • [ilmath]f^{-1}(\mathcal{G})\subseteq\mathcal{A} [/ilmath]


[ilmath]\implies[/ilmath]: [ilmath]f:A\rightarrow B[/ilmath] is [ilmath]\mathcal{A}/\mathcal{F} [/ilmath]-measurable [ilmath]\implies[/ilmath] for some generator [ilmath]\mathcal{G} [/ilmath] of [ilmath]\mathcal{F} [/ilmath] we have [ilmath]\forall S\in\mathcal{G}[f^{-1}(S)\in\mathcal{A}][/ilmath]

  • Let [ilmath]S\in\mathcal{G} [/ilmath] be given
    • Note that [ilmath]\mathcal{G}\subseteq\sigma(\mathcal{G})[/ilmath], so by the implies-subset relation we see [ilmath]S\in\mathcal{G}\implies S\in\sigma(\mathcal{G})[/ilmath]
    • By the definition of [ilmath]\mathcal{A}/\mathcal{F} [/ilmath]-measurable:
      • [ilmath]\forall S\in F[f^{-1}(S)\in\mathcal{A}][/ilmath]
    • Thus [ilmath]S\in\mathcal{G}\implies S\in\sigma(\mathcal{G})=\mathcal{F}[/ilmath]
      • But as we've just seen, if [ilmath]S\in\mathcal{F} [/ilmath] then [ilmath]f^{-1}(S)\in\mathcal{A} [/ilmath]
  • So [ilmath]f^{-1}(S)\in\mathcal{A} [/ilmath]

This completes the proof


TODO: See ref[2] page 6, also lemma 7.2 in[1]


  1. Thus [ilmath]\mathcal{F}=\sigma(\mathcal{G})[/ilmath]


  1. 1.0 1.1 Measures, Integrals and Martingales - René L. Schilling
  2. 2.0 2.1 Probability and Stochastics - Erhan Cinlar