In my mathematical statistics lecture notes, there is a theorem which states:
$$\begin{equation} P\{F(X) \leq F(x)\} = F(x) ,\forall x \in \mathbb{R} \tag{1} \end{equation}$$
but it's not clear to me how I should interpret $F(X)$ here. If we consider that $F$ is defined as:
$$\begin{equation} F(x)=P(X \leq x) \tag{2} \end{equation}$$
then it appears to me that the only logical interpretation of $F(X)$ is:
$$\begin{equation} F(X)=P(X \leq X)=1 \tag{3} \end{equation}$$
but then the theorem statement would be equivalent to:
$$\begin{equation} P\{F(X) \leq F(x)\} = P\{1 \leq F(x)\} = 0 \tag{*} \end{equation}$$
which is absolutely useless. Might there be a sensible interpretation of $F(X)$ which I'm missing?
We have $X$ a real random variable, i.e. $X:\Omega\to\Bbb R$.
Then $F(X)=F\circ X=\omega\mapsto F(X(\omega))$, and for a fixed $x\in\Bbb R$, we have $$(F(X)\le F(x))\ =\ \{\omega:F(X(\omega))\le F(x)\} $$