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

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 $$f_n (x) =n \chi _{(0,\frac{1}{n})}$$

Your answer for the second part is correct. For the first one take $$f_n(x)=\frac 1 {\sqrt n} \chi_{(n,\infty)}$$.
• @amirbahadory Some people write $I_A$ for $\chi_A$. Anyway, I changed it to $\chi$. – Kavi Rama Murthy Jan 11 at 12:29
• I find $\int f_{n} d m=0$ .is this true ? – amir bahadory Jan 11 at 13:56