April 4, 2025
Integration is a linear functional. The integral of a
constant times a function is the constant times the intergral of the
function and the integral of a sum of functions is the sum of the
integrals. The Riemann integral of a continuous function on a finite
interval is defined as a limit:
\int_a^b f(x)\,dx = \lim_{\Delta x\to 0} \sum_j f(x_j^*)\,Delta x_j
where a = x_0 < \cdots < x_n =
b, \Delta x_j = x_{j+1} - x_j,
and x_j^*\in [x_j, x_{j+1}]. The
precise definition of the limit is a bit complicated and it is not
trivial to show it exists and is unique.
A function from a set S to the real numbers, f\colon S\to\boldsymbol{{{R}}}, associates each s\in S with a unique f(s)\in\boldsymbol{{{R}}}. A (finitely additive) measure is a function from the set of all subsets of S, power set of S, \mathcal{P}(S) = \{A\subseteq S\}, to the real numbers \phi\colon\mathcal{P}(S)\to\boldsymbol{{{R}}} that satisfies \phi(\emptyset) = 0 and \phi(A\cup B) = \phi(A) + \phi(B) - \phi(A\cap B). The measure of the empty set is 0 and measures don’t count things twice.
If A and B are sets then the set exponential B^A is the set of all functions from A to B, B^A = \{f\colon B\to A.
Denote the bounded functions on S by B(S) = \{f\colon S\to\boldsymbol{{{R}}}\mid \|f\| = \sup_{s\in S} |f(s)| < \infty\}. The dual of B(S) is the set of linear functionals on B(S), B(S)^* = \{F\colon B(S)\to\boldsymbol{{{R}}}\mid F\text{ is linear}\}. Every F\in B(S)^* determines a measure \phi(A) = F(1_A) where 1_A\colon S\to\boldsymbol{{{R}}} is the characteristic function 1_A(s) = 1 if s\in A and 1_A(s) = 0 if s\not\in A.
Exercise. Show \phi is a measure.
Hint. Use 1_\emptyset = 0 and 1_{A\cup B} = 1_A + 1_B - 1_{A\cap B}.
The integral of f with respect to \phi is \int_S f\,d\phi = F(f).