First version:
This is from an old script of my professor:
Let $f_n$ be a sequence of integrable functions. Let $f$ be a measurable function such that $\lim_{n\to\infty} f_n(\omega) = f(\omega)$ $P$-almost everywhere. Let $g\geq 0$ be a positive function with the property $\int g\,dP<\infty$ such that $|f_n(\omega)|\leq g(\omega)$ $P$-almost everywhere. Then $f$ is integrable and it is true that $\lim_{n\to\infty}\int_\Omega f_n\,dP=\int_\Omega f\,dP$
Second version:
I read this one on Wikipedia:
Let $f_n$ denote a sequence of real-valued measurable functions on a measure space $(\Omega ,\mathcal{A},P)$. Assume that the sequence converges pointwise to a function $f$ and is dominated by some integrable function $g$ in the sense that $|f_n(x)| \leq g(x)$ for all $x\in \Omega$. Then the limiting function $f$ is integrable and $\lim_{n\to\infty}\int_\Omega f_n\,dP=\int_\Omega f\,dP$
The difference here is that $f_n$ are real-valued measurable functions (not integrable as in the version above). Are these versions still equivalent?
Thanks for anyone who enlightens me.