can limit and integral be interchanged for sequence dominated by non-integrable function

Question

I want to build a better intuition behind usage of DCT.

Dominated convergence theorem gives sufficient condition for interchanging limit and integral operations:

given:

1. $f_n \xrightarrow{a.e.} f$

2. there is a function $g$ that dominates $f_n$: $|f_n(x)| < g(x)$ for all $n$

3. g is integrable: $\int g < +\infty$

then $\lim (\int f_n) = \int (\lim f_n)$

I explicitly split conditions "2" and "3", while in textbooks they are often combined into a single one ($f_n$ dominated by integrable function $g$)

I'm wondering whether $\lim (\int f_n) = \int (\lim f_n)$ (allowing both side being equal $+\infty$) if we require just conditions "1" and "2".

Answer 1

The theorem is not true if $g$ is not integrable.

Take the sequence $f_n:=1_{(n,n+1)} \to 0$

and $g(x)=1$ on the real line with the Lebesgue measure.