Measure Theory: What does a.e. (almost everywhere) mean

Source: Elements of Integration by Professor Bartle

Students studying Mathematical Analysis, Advanced Calculus, or probability would sooner or later come across the term a.e. or “almost everywhere”.

In layman’s terms, it means that the proposition (in the given context) holds for all cases except for a certain subset which is very small. For instance, if f(x)=0 for all x, and g(x)=0 for all nonzero x, but g(0)=1, the function f and g would be equal almost everywhere.

For formally, a certain proposition holds \mu-almost everywhere if there exists a subset N\in \mathbf{X} with \mu (N)=0 such that the proposition holds on the complement of N. \mu is a measure defined on the measure space \mathbf{X}, which is discussed in a previous blog post: What is a Measure.

Two functions f, g are said to be equal \mu-almost everywhere when f(x)=g(x) when x\notin N, for some N\in X with \mu (N)=0. In this case we would often write f=g, \mu-a.e.

Similarly, this notation can be used in the case of convergence, for example f=\lim f_n, \mu-a.e.

The idea of “almost everywhere” is useful in the theory of integration, as there is a famous Theorem called “Lebesgue criterion for Riemann integrability”.

(From Wikipedia)

A function on a compact interval [ab] is Riemann integrable if and only if it is bounded and continuous almost everywhere (the set of its points of discontinuity has measure zero, in the sense of Lebesgue measure). This is known as the Lebesgue’s integrability condition or Lebesgue’s criterion for Riemann integrability or the Riemann—Lebesgue theorem.[4] The criterion has nothing to do with the Lebesgue integral. It is due to Lebesgue and uses his measure zero, but makes use of neither Lebesgue’s general measure or integral.

Reference book:

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