Measures

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 x_j-1 ≤ ξ_j ≤ x_j for each j, and the limit is taken over increasingly fine partitions a = x₀ < x₁ < … < x_n= 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 f_n is Riemann integrable for each n, and if f_n(x) → f(x) for every x, a ≤x ≤b, it is entirely possible that f is not Riemann integrable. (For instance, take a = 0 and b=1, and set f_n(x) = 1 if x = m/n! for some integer m, and f_n(x) = 0 otherwise. Then each f_n is Riemann integrable, and f_n 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.