Measures

• Andrew Browder
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

In Chapter 5 we defined the Riemann integral of a real function f over a bounded interval [a, b] by
$$\int_a^b {f\left( x \right)} dx = \lim \sum\limits_{j = 1}^n {f\left( {\xi _j } \right)} \left( {x_j - x_{j - 1} } \right),$$
where xj-1 ≤ ξjxj for each j, and the limit is taken over increasingly fine partitions a = x0 < x1 < … < xn= b of the interval. We found that this limit existed whenever f was continuous on [a,b], in fact, whenever f was bounded, with a set of discontinuities D which was “small,” in the sense that for any ε > 0, there existed a finite collection of open intervals $$\left\{ {\left( {a_k ,b_k } \right):k = 1, \ldots r} \right\}$$ such that
$$D \subset \mathop \cup \limits_{k = 1}^r \left( {a_k ,b_k } \right)and\sum\limits_{k = 1}^r {\left( {b_k - a_k } \right)} < \epsilon$$
This is a fairly rich class of functions, including as it does not only every continuous function, but also some functions which have infinitely many, even uncountably many, discontinuities (recall that the Cantor set is small in the above sense.) However, the class of Riemann integrable functions does have at least one glaring weakness: it is not stable under pointwise convergence. That is, if fn is Riemann integrable for each n, and if fn(x) → f(x) for every x, axb, it is entirely possible that f is not Riemann integrable. (For instance, take a = 0 and b=1, and set fn(x) = 1 if x = m/n! for some integer m, and fn(x) = 0 otherwise. Then each fn is Riemann integrable, and fn converges pointwise to the function f, where f(x) = 1 if x is rational, and f(x) = 0 when x is irrational. We have seen that f is not Riemann integrable.

Keywords

Open Interval Finite Union Outer Measure Countable Additivity Disjoint Sequence
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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