Past exam of the options II course of the University of Oxford / 2008 / D / D3 / a / ii / Solution

No. Take $f_n=1_{[n,n+1]}$. Each is nonnegative and integrable, and each integral is one. At every fixed real point the functions eventually vanish, so the pointwise limit is zero and its integral is zero. Monotonicity or another convergence hypothesis is essential.