Is the following true: if $X = Y$ in distribution, then $XI_{\{|X| < a\}} = YI_{\{|Y| < a\}}$ in distribution, where $I$ denotes the indicator function on the given set, and where $a$ is a fixed constant
I have to prove that $\mathbb{P}\{XI_{\{|X| < a\}} \in A\} = \mathbb{P}\{YI_{\{|Y| < a\}} \in A\}$ for any Borel set of the reals, but I honestly have no clue how to prove it. The statement seems intuitively true.
Assuming $X$ and $Y$ are real-valued, take any $b\in \mathbb{R}$; then it is enough to show that $$ \mathbb{P}(X I_{|X|<a} \leq b) = \mathbb{P}(Y I_{|Y|<a} \leq b) . $$ We have $$ \mathbb{P}(X I_{|X|<a} \leq b) = \mathbb{P}(-a \leq X \leq a, X \leq b ) + \mathbb{P}(|X| > a) \chi_{\{b\geq 0\}} = \\ \mathbb{P}(-a \leq X \leq \min\{a,b\}) + \mathbb{P}(|X| > a) \chi_{\{b\geq 0\}} = \\ \mathbb{P}(-a \leq Y \leq \min\{a,b\}) + \mathbb{P}(|Y| > a) \chi_{\{b\geq 0\}} = \\ \mathbb{P}(Y I_{|Y|<a} \leq b) $$ since $X$ and $Y$ have the same distribution.