Is this function measurable?

Question

Let $(B, \Sigma_B)$ and $(C, \Sigma_C)$ be standard Borel spaces and let $\mu$ be a sub-probability measure on $C$.

Given $Y\in \Sigma_{B\times C}$, I would like to use the following function: $$ f: B \to [0, 1] \\ b \mapsto \mu(\{c \in C \mid (b, c) \in Y\}) $$

But I am stuck trying to prove that it is well-defined (i.e. the set in brackets is measurable) and measurable.

I suppose first you need to prove that $\Sigma_{B\times C}$ is the product sigma-algebra from $\Sigma_B$ and $\Sigma_C$. (Maybe that is the defintion? I don't know this notation.) Then it follows from the definition of "product sigma-algebra" ... if $Y$ is one of the generators of the product sigma-agebra, then your set is well-defined; and the collection of $Y$ such that your set is well-defined is a sigma-algebra. By the way: you also need to define "sub-probability measure". — Gerald Edgar, Commented Mar 20, 2018 at 18:36

Fedor Petrov · Accepted Answer · 2018-03-20 15:25:02Z

2

Yes, it is a part of Fubini theorem for the characteristic function of $Y$.

answered Mar 20, 2018 at 15:25

Fedor Petrov

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