Let $f:\mathbb R\rightarrow[0,\infty)$ be Lebesgue measurable. Is it true in general that the function:
$$\mu(E):=\int_E f$$
is a measure? The right hand side is the Lebesgue integral. I am having trouble proving the property of countable disjoint union. Suppose that $E_n$ ($n\geq1$) are disjoint sets, then:
$$\int_{\bigcup_{n\geq 1}E_n}f\quad\mathop{=}^{?}\quad\sum_{n\geq1}\int_{E_n}f$$
I know this equality is true for finite disjoint unions (using the linearity of the integral and the fact that $\chi_{\sum E_n}=\sum\chi_{E_n}$), but is it also true for countably infinite disjoint unions? (the linearity would fail in this case!).
Best Answer
If $f$ is Lebesgue integrable, then yes, $\mu$ is an absolutely continuous measure with respect to Lebesgue measure.
Monotone Convergence of the indicator functions gives you the property of countable disjoint unions.