Research proof index

Suppose that, for every error threshold $\delta>0$ and every measure tolerance $\varepsilon>0$, one can choose $N_0$ so that the set where $\sup_{N>N_0}\lVert f(x)-F_N(x)\rVert$ exceeds $\delta$ has measure at most $\varepsilon$. Then $F_N(x)$ converges to $f(x)$ for almost every $x$.