Convergence Almost Everywhere
Given a sequence of extended real-valued functions on a measure space , we say that converges a.e. to the function if there is a set with so that for all . Similarly, we say that the sequence is Cauchy a.e. if there exists a set of measure zero so that is a Cauchy sequence of real numbers for all . That is, given and there is some natural number depending on and so that whenever we have
Because the real numbers form a complete metric space, being Cauchy and being convergent are equivalent — a sequence of finite real numbers is convergent if and only if it is Cauchy, and a similar thing happens here. If a sequence of finite-valued functions is convergent a.e., then converges to away from a set of measure zero. Each of these sequences is thus Cauchy, and so is Cauchy almost everywhere. On the other hand, if is Cauchy a.e. then the sequences are Cauchy away from a set of measure zero, and these sequences then converge.
We can also define what it means for a sequence of functions to converge uniformly almost everywhere. That is, there is some set of measure zero so that for every we can find a natural number so that for all and we have . The uniformity means that is independent of , but if we choose a different negligible we may have to choose different values of to get the desired control on the sequence.
As it happens, the topology defined by uniform a.e. convergence comes from a norm: the essential supremum; using this notion of convergence makes the algebra of essentially bounded measurable functions on a measure space into a normed vector space. Indeed, we can check what it means for a sequence of functions to converge to under the essential supremum norm — for any there is some so that for all we have . Unpacking the definition of the essential supremum, this means that there is some measurable set with measure zero so that for all , which is exactly what we said for uniform a.e. convergence above.
We can also turn around and define what it means for a sequence to be uniformly Cauchy almost everywhere — for any there is some so that for all we have . Unpacking again, there is some measurable set so that for all . It’s straightforward to check that a sequence that converges uniformly a.e. is uniformly Cauchy a.e., and vice versa. That is, the topology defined by the essential supremum norm is complete, and the algebra of essentially bounded measurable functions on a measure space is a Banach space.
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