Let $\mu$ be a positive measure on $X$. A sequence $\{f_n\}$ of complex measurable functions on $X$ is said to \emph{converge in measure} to the measurable function $f$ if to every $\epsilon > 0$ there corresponds an $N$ such that \[ \mu(\{x : \lvert f_n(x) - f(x) \rvert > \epsilon \}) N$. Assume $\mu(X) < \infty$ and prove the following statements: \begin{enumerate}[label=(\alph*)] \item If $f_n(x) \to f(x)$ a.e., then $f_n \to f$ in measure. \item If $f_n \in L^p(\mu)$ and $\lVert f_n - f \rVert_p \to 0$, then $f_n \to f$ in measure; here $1 \le p \le \infty$. \item If $f_n \to f$ in measure, then $\{f_n\}$ has a subsequence which converges to $f$ a.e. \end{enumerate} Invetigate the converses of $(a)$ and $(b)$. What happens to $(a)$, $(b)$, and $(c)$ if $\mu(X) = \infty$, for instance, if $\mu$ is Lebesgue measure on $R^1$?

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Exercise 3.18

Let $μ$ be a positive measure on $X$ . A sequence ${f_{n}}$ of complex measurable functions on $X$ is said to converge in measure to the measurable function $f$ if to every $𝜖 > 0$ there corresponds an $N$ such that

μ ({x : | f_{n} (x) - f (x) | > 𝜖}) < 𝜖

for all $n > N$ . Assume $μ (X) < \infty$ and prove the following statements:

(a): If $f_{n} (x) \to f (x)$ a.e., then $f_{n} \to f$ in measure.
(b): If $f_{n} \in L^{p} (μ)$ and $∥ f_{n} - f ∥_{p} \to 0$ , then $f_{n} \to f$ in measure; here $1 \leq p \leq \infty$ .
(c): If $f_{n} \to f$ in measure, then ${f_{n}}$ has a subsequence which converges to $f$ a.e.

Invetigate the converses of $(a)$ and $(b)$ . What happens to $(a)$ , $(b)$ , and $(c)$ if $μ (X) = \infty$ , for instance, if $μ$ is Lebesgue measure on $R^{1}$ ?

Exercise 3.18

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