I have a doubt regarding the Lebesgue monotone convergence theorem.
The version that I know is the following from Wikipedia, requiring, in particular, $\{f_k(x)\}$ monotone increasing and $f_k(x)\geq 0$ $\forall k, x$.
Do you know other versions of the theorem? For example, do you know a version of the theorem (i) requiring $\{f_k(x\})$ monotone decreasing and (ii) allowing for $f_k(x)$ positive and negative across $k,x$?
I'm asking this because I don't understand which monotone convergence theorem is applied in van der Vaart "Asymptotic Theory" proof of Theorem 5.14.
If $f_n$ is monotone decreasing, then $g_n=f_1-f_n$ is monotone increasing and always positive, regardless of the sign of $f_n$ (note that monotonicity allows one change of sign, anyway). So you can apply standard MCT to prove that $g_n \to f_1-f$.