Exercise 1.4.45 (Dominated convergence implies $L^1$-convergence)

Question

Under the hypotheses of dominated convergence theorem (Theorem 1.4.48), establish also that
    $$\lim_{n	o\infty}  \|f_n-f\|_{L^1} = 0 $$

Elu Thingol · Accepted Answer

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ewcommand{\intx}[1]{\int_{X} #1 \mathop{}\!\mathrm{d}{\mu}}$
We look at whether the function $\|f_n-f\|$ is a candidate for the Dominated Convergence Theorem (Theorem 1.4.48). Since $|f_n|$ are bounded by $G$, so is $f$. We then have
$$\left| f_n - f ight| \leq |f_n| + |f| \leq 2G.$$
By \href{./exercise_1-4-29}{Exercise 1.4.29}, $|f_n-f|$ is a sequence of measurable functions, and it converges to $0$ almost everywhere. Thus,
$$\lim_{n	o\infty} \intx{\left| f_n - f ight|} = \intx{\lim_{n	o\infty}\left| f_n - f ight|} = 0.$$

Exercise 1.4.45 (Dominated convergence implies $L^1$-convergence)

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Exercise 1.4.45 (Dominated convergence implies $L^1$-convergence)

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