What's the intuition behind why the conditional expectation w.r.t. a $\sigma$-algebra allows us to condition to events with zero probability?
For example, let’s say we have two continuous random variables $X$ and $Y$ defined on $(\Omega,\mathcal{F},\mathbb{P})$. We define the conditional probability density function of $Y$ given $X=x_0$, as $$f_{Y|X}(y,x_0)=\frac{f_{XY}(x_0,y)}{f_X(x_0)}.$$
Even though in this definition $f_X$ has to be strictly greater than $0$, we are still conditioning to an event with probability zero $\mathbb{P}(X=x_0)=0$, as $X$ is a continuous r.v.
I understand that the concept of conditional expectation of a random variable (and conditional probability of an event) to a $\sigma$-algebra was introduced in order to deal with conditioning to events with probability zero, but I haven’t quite grasped the intuition behind this.