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. 



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


