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I know that $\sigma(\mathcal{G})$ for a family of sets $\mathcal{G}$ is the smallest $\sigma$-algebra which contains $\mathcal{G}$.

I'm reading some online notes, and ran into the term $\sigma(T)$ where $T$ is a measurable map from one measurable space $(X,\mathcal{A}$) into another $(X', \mathcal{A}')$.

So what exactly do they mean by a $\sigma$-algebra generated by a $\textit{function}$ as opposed to a set?

iikao sero
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  • Maybe this link can be useful: http://math.stackexchange.com/questions/23851/question-on-the-notion-of-a-sigma-algebra-generated-by-a-function – Kolmin Sep 14 '15 at 15:06

1 Answers1

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$$\sigma(T) = \sigma\{ T^{-1}(B) : B \in \mathcal A'\}.$$ It is the minimal $\sigma$-algebra with the property that the preimage of every measurable set is measurable.

Umberto P.
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