Let $X,Y$ be two random variables on $(\Omega,\mathcal{F},P)$ taking values in $\mathbb{R}^d, \mathcal{B}(\mathbb{R}^d)$ such that $X$ is independent of $\mathcal{G} \subset \mathcal{F}$ and $Y$ is $\mathcal{G}$-measurable. I am trying to show that for a borel set $\Gamma \in \mathcal{B}(\mathbb{R}^d)$
$$ P[X+Y \in \Gamma \mid Y=y]=P[X+y \in \Gamma] \text{ for } PY^{-1} \text{-a.s.} y \in \mathbb{R}^d $$ Can I write that
$ P[X+Y \in \Gamma \mid Y=y]=\frac{P[X+Y \in \Gamma, Y=y]}{P[Y=y]}= \frac{P[X+y \in \Gamma, Y=y]}{P[Y=y]}= \frac{P[X+y \in \Gamma] P[ Y=y]}{P[Y=y]} = P[X+y \in \Gamma]$
The problem is I cannot use this definition of conditonal probability if $P[Y=y]$ has measure zero. how could i resolve this issue? I am studying the markov property.
It is direct that: $$P(X+Y\in\Gamma\mid Y=y)=P(X+y\in\Gamma\mid Y=y)$$
From $X$ being independent of $\mathcal G$ and $Y$ being $\mathcal G$-measurable it follows that $X$ and $Y$ are independent so that also: $$P(X+y\in\Gamma\mid Y=y)=P(X+y\in\Gamma)$$