Taken off University of South Carolina's Analysis PhD qualifying exam, January 1999
PROBLEM
Let $(X,\Sigma,\mu)$ be an arbitrary complete measure space and $L_0$ be the collection of all $\mu$-measurable functions from $X\rightarrow \mathbb{R}$, and$\{f_n\}$ be a sequence of functions from $L_0$ that converge almost everywhere to $f_0\in L_0$. Does $f_n\rightarrow f_0$ almost uniformly? (note that $X$ need not be finite).
COUNTEREXAMPLE
Egorov's theorem only holds for finite complete measure spaces. For example, consider $f_n=\chi_{\mathbb{R}\backslash [-n,n]}$ where $\chi$ is the indicator or characteristic function. $f_n\rightarrow 0$ everywhere but for all subsets $K$, $\mu(\mathbb{R}\backslash K)<\epsilon$. Hence $f_n$ does not converge to 0 uniformly.
PROBLEM
Let $(X,\Sigma,\mu)$ be an arbitrary complete measure space and $L_0$ be the collection of all $\mu$-measurable functions from $X\rightarrow \mathbb{R}$, and$\{f_n\}$ be a sequence of functions from $L_0$ that converge almost everywhere to $f_0\in L_0$. Does $f_n\rightarrow f_0$ almost uniformly? (note that $X$ need not be finite).
COUNTEREXAMPLE
Egorov's theorem only holds for finite complete measure spaces. For example, consider $f_n=\chi_{\mathbb{R}\backslash [-n,n]}$ where $\chi$ is the indicator or characteristic function. $f_n\rightarrow 0$ everywhere but for all subsets $K$, $\mu(\mathbb{R}\backslash K)<\epsilon$. Hence $f_n$ does not converge to 0 uniformly.
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