For a continuous RV $X$ the general transformation to a RV $Y$ is given by:
$ P_y(y) = P(Y≤y) = P(f(X)≤y) = P(X \in \{x|f(x) ≤y\}) $
Could someone explain the meaning opf the final term please? This notation comes from the book Machine Learning: A Probabilistic perspective by Murphy on page 50.
When the functions are monotonic and invertible the no0tation becomes
$P_y(y) = P(Y≤y) = P(f(X)≤y) = P(X ≤ f^{-1}(y)) = P_x(f^{-1}(Y))$
which I understand.
That line can be read as "the probability that $X$ is a value such that it's image (under $f$) is less than or equal to $y$" If our transformation isn't invertible, this is all the further we are able to simplify, unlike invertible transformations that can be further written as $P(X \le f^{-1} (y))$.
A good example would be to use the space $(-\infty, \infty)$ for $X$ and the transformation $Y = f(X) = X^2$. This transformation is not one-to-one and not invertible. In this case $f^{-1}(y)$ doesn't have a definite meaning.