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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.

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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.

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  • $\begingroup$ +1 What sequence are you applying Monotone Convergence Theorem to? $\endgroup$ – Pascal's Wager 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

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