Consider two nonnegative random variables, $X$ and $Y$.
Assume that their PDF, i.e., $f_X(x)$ and $f_Y(y)$ are given.
Assume that an arbitrary function $g:X\times Y\to\mathbb{R_{\ge0}}$ is given.
However, we don't know the PDF of $Z=g(X,Y)$.
Then, for a given constant value $\gamma>0$, I want to derive $\Pr\left[g(X,Y) \le \gamma \right]$.
Let us define a function $h$ such that $$g(x,y)\le\gamma \quad \Leftrightarrow \quad x\le h(y,\gamma),$$ where $\Leftrightarrow$ denotes the if and only if symbol. Then, is the following equation correct? $$\Pr\left[f(X,Y)\le\gamma\right] = \int_0^\infty \left( \int_0^{h(y,\gamma)}f_{X}(x)\,\text{d}x\right)f_Y(y)\,\text{d}y.$$