# Pointwise convergence and convergence of integrals implies $L^1$ convergence

From Donald Cohn's Measure Theory, section 2.4, exercise 10.

Let $$(X, A, \mu)$$ be a measure space, and let $$f$$ and $$f_1, f_2, \dots$$ be non-negative functions that belong to $$L^1(X, A, \mu, R)$$ and satisfy

(i) $$\{f_n\}_n$$ converges to $$f$$ almost everywhere;

(ii) $$\int fd\mu = \lim_n\int f_nd\mu$$.

Show that $$\lim_n\int |f_n - f|d\mu = 0$$.

I let $$f_n = 1/n$$ on $$[n, n+1]$$ and $$0$$ elsewhere, and let $$f=0$$, then all the convergence theorems in that section (dominated convergence, and monotone convergence) failed.

• What have you tried? There basically only 3 convergence theorems for Lebesgue integral. Jan 23 '17 at 23:52
• I let f_n = 1/n on [n, n+1] and 0 elsewhere, then all the convergence theorems in that section (dominated convergence, and monotone convergence) failed. Jan 23 '17 at 23:55
• They converge to f = 0. |f_n - f| integrates to 1/n which converges to 0. Jan 24 '17 at 0:09
• What's the third convergence theorem you mentioned? Jan 24 '17 at 0:11

Fatou's Lemma yields $$2\int f = \int \liminf_{n\to\infty} \left ( f_n + f - |f_n - f| \right ) \le \int f + \int f + \liminf_{n\to\infty} \left ( - \int |f_n - f| \right ).$$
Let $F_n = f_n-|f_n-f|$. Then $F_n$ converges pointwise to $f$ and we have by the reverse triangle inequality $|F_n| = |f_n-|f_n-f|| = ||f_n|-|f_n-f||\leq |f|$. Hence by the dominated convergence theorem $\lim_n \int F_n = \int f$. But then $$\lim_n \int |f_n-f| = \lim_n \int (|f_n-f|-f_n+f_n) = -\lim_n \int F_n + \lim_n\int f_n = 0$$ since $\lim_n \int f_n = \int f$.