Suppose $X_1$ and $X_2$ are iid random variables. I want to determine $P(X_1=X_2)$.
If they are integer-valued random variables, then $$P(X_1=X_2) = \sum_{i \in \mathbb{Z}} P_{X_1,X_2}(i,i) = \sum_{i \in \mathbb{Z}} P_{X_1}(i)^2. $$
If they are continuous random variables, then $$P(X_1=X_2) = \int_{x \in \mathbb{R}} f_{X_1,X_2}(x,x) dx = \int_{x \in \mathbb{R}} f_{X_1}(x)^2 dx. $$ But when $X_1$ and $X_2$ are uniformly distributed over $[0,1)$, $$P(X_1=X_2) = \int_{x \in \mathbb{R}} f_{X_1}(x)^2 dx = \int_{x \in [0,1)} 1 dx = 1. $$ Intuitively it is not possible, since $P(X_1\neq X_2) > 0$. So is there some mistake I have made? Thanks!