What does this notation mean in the context of probability theory?

Question

Let X be a real random variable with the density $f_X$ with respect to lebesgue measure given as…,which means $$P_X=f_X \cdot m$$ is a uniform distribution on the interval (0,1). What does it mean when $P_X$ is written that way? Is it a pushforward measure of a lebesgue measure or something else?(m is lebesgue measure)

Thanks in advance

Answer 1

$P_X(A)=P(X^{-1}(A))$ for any Borel set $A$. $P_X$ is a probability measure on Borel sets of $\mathbb R$. The given equation says $P_X(A)=\int_A f_Xdm$ where $m$ is Lebesgue measure on $\mathbb R$.