Let $(X, \mathcal{S}, \mu)$ a measure space and $f_1, f_2, \dots$ a monotone sequence of $\mathcal{S}$-measurable functions. Define $f:= \lim_n f_n$. If $\int f_1^- < \infty$, then
$$\lim_n \int f_n d \mu= \int f d \mu$$
Attempt: We may assume $f_1^-$ is real valued. We have
$$\int (f_n + f_1^-) = \int f_n^+ + \int (f_1^- -f_n^-)$$
$$\nearrow \int f^+ + \int (f_1^- – f^-) = \int (f+ f_1^-)$$
by the classical monotone convergence theorem and linearity of the integral of functions $X \to [0, \infty]$
Since $$\int f_1^- < \infty$$
it follows that $$\int f_n \nearrow \int f$$
Is this correct?
Best Answer
It is correct but I think should state clearly why $\int f_n$ makes sense. We have $0 \leq f_n^{-} \leq f_1^{-}$ so $\int f_n^{-} <\infty$ and $\int f_n$ is defined. Now your last step is justified.