Let $X,Y$ be two independent random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and let $g,f :\mathbb{R}^2 \to \mathbb{R}$ be continuos functions. Finally let $A = Y(\Omega)$ be the image of $Y$. Now assume that for any $a \in A$ we have the distributional equality $$\mathbb{P}^{g(X,Y) \mid Y = a} = \mathbb{P}^{f(X,Y) \mid Y = a}.$$
Does this then imply that $g(X,Y)$ and $f(X,Y)$ already have the same distribution? Is this an application of the law of total expectation?
And further, could we use the same argument for the following problem: Assume that the random variable $[g(X,Y)\mid Y=a]$, i.e. a random variable with distribution $\mathbb{P}^{g(X,Y) \mid Y = a}$ has the same distribution for any $a \in A$, for example $$[g(X,Y)\mid Y=a] \sim \mathcal{N}(0,1)$$ could we then infer that also $g(X,Y) \sim \mathcal{N}(0,1)$?
Thanks in advance.
Yes because for any $B \subset \mathbb{R}$, $$P(g(X, Y) \in B) = \int P(Y \in da)P(g(X, Y) \in B \mid Y = a).$$