Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 7 down vote accepted

Since each $f_n$ satisfies $|f_n(\omega)| \leq g(\omega)$ almost everywhere and $g$ is integrable, it follows from the monotonicity of the integral that each $f_n$ is integrable, so the second version follows from the first.

The first version follows from the second version by throwing away a null-set $N$ containing all points where $f_n$ doesn't converge to $g$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.