Find a sequence $\left\{f_{n}\right\}$ of Borel measurable functions on $\mathbb{R}$ which...

Question

Find a sequence $\left\{f_{n}\right\}$ of Borel measurable functions on $\mathbb{R}$ which decreases uniformly to zero on $\mathbb{R},$ but $\int f_{n} d m=\infty$ for all $n .$ Also, find a sequence $\left\{g_{n}\right\}$ of Borel measurable functions on $[0,1]$ such that $g_{n} \rightarrow 0$ pointwise but $\int g_{n} d m=1$ for all $n$.I think we can consider $f_n (x) =n \chi _{(0,\frac{1}{n})}$

Answer 1

Your answer for the second part is correct. For the first one take $f_n(x)=\frac 1 {\sqrt n} \chi_{(n,\infty)}$.