Does convergence theorem holds for decreasing function?

Question

I was wondering there is no convergence monotone theorem for decreasing function, i.e. if $\varphi _n\searrow f$ and $f\geq 0$, why it may not hold that $$\lim_{n\to \infty }\int_X \varphi _n=\int_Xf\ \ ?$$

So, I saw this link, but they don't give an example of such sequence.

$\chi_{(n,\infty)}$ on the real line. – Kavi Rama Murthy Jan 20 '21 at 10:21 — Kavi Rama Murthy, Jan 20 '21 at 10:21

score 1 · Answer 1 · answered Jan 20 '21 at 10:22

1

Take for example $$\varphi _n(x)=\infty \boldsymbol 1_{(0,\frac{1}{n})}.$$

Then $\varphi _n\searrow f$, but $\int \varphi _n=\infty $ for all $n$. Notice that this theorem holds if $\int \varphi _n<\infty $ for some $n$ (by DCT).

answered Jan 20 '21 at 10:22

Surb

57,262
11
68
119

Does convergence theorem holds for decreasing function?

1 Answers1