I need help in this problem:
Construct a sequence of non-negative Borel measurable functions $f_n : [0, 1] → [0, +∞)$ such that $\lim_{n→∞}\int_{[0, 1]} f_n(x) dx = 0$ but for every $x ∈ [0, 1]$, one has $\sup_n f_n(x) = +∞$.
Dose $f_n (x)=n x^{n^2}$ work? I can not think of another one!
Any help would be appreciated . Thanks!
Divide the interval $[0,1]$ into $n^2$ sub-intervals of length $1/n^2$. For each subinterval, you can consider the function that has the value $n$ on this subinterval and the value zero everywhere else. Its integral is $n\cdot 1/n^2=1/n$. List all such functions for $n=1$, then $n=2$, and so on. The resulting sequence has the desired property.