Say I have two continues variables $X$ and $Y$ with joint PDF $f(x, y)$, to calculate the probability $P(X=Y)$, I have to take double integral over the "line area", that is $0$.
Can I think $P(X=Y)$ as the sum of all $f(x, y)$ where $(x, y)$ subject to $x=y$, then use line integral to sum all $f(x, y)$ up: $\int_L f(x,y) \, \mathrm{d}\sigma$ where $L$ is $x=y$. The answer is obviously wrong, so my question is, why I can't think like this?
I guess it is something related with the differential element $\mathrm{d}\sigma \in \mathbb{R}$ and $\mathrm{d}x\mathrm{d}y \in \mathbb{R}^2$?
The line $L:x=y$ in $\mathbb{R}^2$ has Lebesgue measure zero, so as long as $x,y$ are continuous variables, $P(X=Y)$ is always zero. $$P(X=Y)=\int_{-\infty}^{\infty}\int_y^y f(x,y)\mathrm{d}x\mathrm{d}y=\int_{-\infty}^{\infty}0\cdot \mathrm{d}y=0$$ Or we have $$P(X=Y)=\int_L f(x,y) \, (0\cdot\mathrm{d}\sigma)=0$$