Let $f_n:E \to [-\infty, \infty]$ be a sequence of (Lebesgue) integrable functions converging pointwise to a function $f$. Assume, additionally, that the sum $\int_E|f_2-f_1|+\int_E|f_3-f_2|+...$ converges to a real number.

My question is, must we necessarily have $\lim_{n \to \infty} \int_Ef_n= \int_Ef$? If not, what would be a counterexample?

I think the statement is probably true, and my thought was to try using the Dominated Convergence Theorem or perhaps even the Generalized Dominated Convergence Theorem. I was thinking of trying to apply DCT to $f_n-f$, or $f_1-f_n$, or something of the like, but I don't know what would work best.

Since the above sum converges, I have tried using the triangle inequality to see that $\int_E|f_n-f_1|< \infty$ for each $n$, but I don't know how this helps. In fact, I think it is true even without the condition that the sum converges, since we are assuming $f_n$ and $f_1$ are both integrable and thus the difference must be integrable.


Note that, by monotone convergence, for any $n$, $$ \int_E |f_n-f|=\int_E \left|\sum_{k=n}^{\infty} f_k-f_{k+1}\right|\leq \int_E \sum_{k=n}^{\infty} |f_k-f_{k+1}|=\sum_{k=n}^{\infty}\int_E |f_k-f_{k+1}|, $$

which is the tail of a convergent series and hence, is arbitrarily small for $n$ sufficiently large. However, this implies that $f\in L^1$ and

$$ \left|\int_E f_n-f \right|\leq \int_E |f_n-f|\to 0, $$ implying the desired.

  • $\begingroup$ +1 What sequence are you applying Monotone Convergence Theorem to? $\endgroup$ – Algebraist Dec 7 '19 at 21:01
  • $\begingroup$ $N\mapsto \sum_{k=n}^N |f_k-f_{k+1}|.$ Basically, monotone convergence says that countable sums and integrals always commute when the summands in question are a.e. positive. $\endgroup$ – WoolierThanThou Dec 7 '19 at 21:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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