Past exam of the Probability, statistics and options II course of the University of Oxford / 2008 / D / 3 / a

The monotone convergence theorem states that if $f_n$ are measurable, nonnegative functions on a measure space with $f_n\le f_{n+1}$ almost everywhere and $f_n\uparrow f$ almost everywhere, thenThe equality is valid in $[0,\infty ]$, so finiteness is not required. Changes on null sets do not affect the conclusion.

No. Take $f_n=1_{[n,n+1]}$ on the real line. At every fixed point it eventually vanishes, so $f=0$, while $\int f_n=1$ for every $n$. Thus the limit of the integrals is one and the integral of the pointwise limit is zero. Nonnegativity alone cannot replace monotonicity or an integrable domination hypothesis.