0
$\begingroup$

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

$\endgroup$
0
$\begingroup$

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

$\endgroup$
  • $\begingroup$ @amirbahadory Some people write $I_A$ for $\chi_A$. Anyway, I changed it to $\chi$. $\endgroup$ – Kavi Rama Murthy Jan 11 at 12:29
  • $\begingroup$ I find $\int f_{n} d m=0$ .is this true ? $\endgroup$ – amir bahadory Jan 11 at 13:56
  • $\begingroup$ No, the integral is infinity. $\endgroup$ – Kavi Rama Murthy Jan 11 at 13:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.