According to Wikipedia, regarding Fubini's termwise differentiation theorem, if I change the hypothesis of $f_k$ increasing I have to use a stricter condition like uniform convergence of the series of the $f_k'$ almost everywhere. That means there exists $f:[a,b] \to \mathbb{R}, f(x) = \sum_{k=1}^{\infty} f_k(x), f_k:[a,b] \to \mathbb{R}$ increasing where the convergence of $\sum_{k=1}^{\infty} f_k'(x)$ is not uniform to $f'(x)$ almost everywhere, since this condition is stricter.
This concept is kind of new to me, so: Am I interpreting this correctly? If yes, what would be a counterexample, i.e., an example where $f:[a,b] \to \mathbb{R}, f_k:[a,b] \to \mathbb{R}$ increasing, $f(x) = \sum_{k=1}^{\infty} f_k(x)$ , $f'(x) = \sum_{k=1}^{\infty} f_k'(x)$ almost everywhere but only pointwise and not uniformly.
On $[0,1]$ define $f_k(x)= x^k/k^2$ for $k=1,2,\dots.$ Then each $f_k$ is increasing and $\sum_{k=1}^{\infty} f_k(x)$ converges uniformly on $[0,1].$ But
$$\sum_{k=1}^{\infty} f_k'(x)= \sum_{k=1}^{\infty} x^{k-1}/k$$
does not converge uniformly on $[0,1].$