Measure

Before we are able to understand Riemann-Stieltjes Integral we need to define the notion of a measure.

Definition (Measure)

A measure in the context of 140B is a function

$$m:\{\text{closed, connected intervals}\}\to \mathbb{R}$$

with the following properties:

1. if $E\subseteq F$ then $m(E)\le m(F)$
2. $m(\varnothing )=0$
3. $m([a,c])=m([a,b])+m([b,c])$ where $a<b<c$ The standard measure is $m([a,b])=b-a$.

To link the notes, we use Connected Sets and Closed Intervals.

Example 1

Let $f:\mathbb{R}\to \mathbb{R}$ be increasing and define

$$m([a,b])=f(b)-f(a)$$

If $f(x)=x^3$ then $m([0,1])=1$ and $m([0,2])=8$ and so on.

At this point, pedagogy dictates you should read Riemann-Stieltjes Integral.