# Proof of countable additive property of Lebesgue Integrable functions

I am trying to prove the following

Theorem: Let $$\{A_1, A_2, \cdots \}$$ be a countable disjoint collection of sets in $$\mathbb R$$ and let $$S = \bigcup_{i=1}^\infty A_i$$. Let $$f$$ be defined on $$S$$.

(a) If $$f\in L(S)$$, then $$f\in L(A_i)$$ for each $$i$$ and $$\int_S f = \sum_{i=1}^\infty \int_{A_i} f.$$ (b) If $$f\in L(A_i)$$ for each $$i$$ and if the series in (a) converges, then $$f\in L(S)$$ and the equation in (a) holds.

(a) is easy, but I am not able to prove part (b) as I don't know what result previously proved can be used because $$f$$ is not a sequence.

• (b) is not true. (It is true if $f \geq 0$.) It is easy to construct a function $f$ such that $\int_n^{n+1}f(x)dx=0$ for all $n$ but $f$ is not integrable on the union of the intervals $(n, n+1)$. Jun 29, 2020 at 8:03
• @Thanks for telling. Can you kindly outline a proof if f$\geq$0 ? It would be really helpful as I am struck on it from many days. Jun 29, 2020 at 8:17

Proof of (b) when $$f \geq 0$$. [As already pointed out (b) is false in general].
Let $$f_n= \sum\limits_{i=1}^{n} f\chi_{A_i}$$. Then $$(f_n)$$ is a sequence of non-negative measurable functions increasing to $$f$$. By Monotone Convergence Theorem we have $$\int_S f= \lim \int_A f_n=\lim \sum\limits_{i=1}^{n} \int_{A_i} f$$. Hence $$\int_S f <\infty$$ and $$\int_S f =\sum\limits_{i=1}^{\infty} \int_{A_i} f$$.