What are the problems with the conditional expectation of $\mathbb{E}(X|H)$ when $\mathbb{P}(H)=0$?

Question

Is there a "problem" with the conditional expectation of $X$ given an event $H$ when $H$ is a null set?

For example $X=(X_t)_{t\geq0}$ could be a gaussian Process $\Omega\times\mathbb{R}^+\mapsto\mathbb{R}$ and $H=\{X_T=x\}$ for some $T\in\mathbb{R}^+,x\in\mathbb{R}$.
Problem could be loss of $\mathbb{P}$-a.s. statements or behavior of $\mathbb{E}(X|H)$ that is fundamentally different from the usual behavior of $X$.

If so what are ways to circumvent the problems?

Answer 1

The "problem" in general is that there is no way to define $\mathbb E[X | H]$ when $H$ is an event of probability $0$. But there is a way to define conditional expectations given a continuous random variable: that's not the same thing.