Given a measurable function $f: X \to Y$, if $f$ is injective and measure-preserving that $\mu(f(A))=\mu(A)$ for all subsets $A$ and $\mu$ is a probability measure, can we deduce that $\mu(X)\leq \mu(Y)$?
Do we have $\mu(f(X))=\mu(X)\leq \mu(Y)$ since $f(X)\subset Y$?
The question now very different from the one for which I posted this answer.
$\mu(X)=\mu (f^{-1}(Y))=\mu (Y)$ so you have equality.