# $\int_E|f_2-f_1|+\int_E|f_3-f_2|+…< \infty$ allows us to pass limit under integral sign?

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.
• $N\mapsto \sum_{k=n}^N |f_k-f_{k+1}|.$ Basically, monotone convergence says that countable sums and integrals always commute when the summands in question are a.e. positive. – WoolierThanThou Dec 7 '19 at 21:09