Introduction to measure

Although integration is often introduced in terms of Riemann sums, it is rather clear that Riemann integration only really represents a specific algorithm for approximating volumes based on some intuition that applies to sufficiently familiar-looking functions. In particular, it doesn't really help us in trying to understand what a volume is (the standard example is that it doesn't tell us what the volume of $\mathbb{Q}$ is).

Even without calculus, we can already calculate the volumes of polygonal shapes like lines and trapeziums through basic linear algebra -- and calculus is the science of being able to define volumes of more complicated shapes based on limiting rules from these building blocks. Riemann integration is an example of such a limiting process.

The general idea behind abstract measure theory is that we can define volume on an arbitrary space based on some "building block" sets. Essentially, if we have a set $\mathcal{F}_0$ (called a sigma basis) of shapes whose volumes we can know axiomatically, then we should be able to say that the volume of a countable union of such disjoint shapes is the sum of their volumes and the volume of the difference between a shape and its subshape is the difference between their volumes.

(These axioms look awfully similar to those of probability theory, which is why measure theory appears so much in probability. Countable unions in the definition of sigma algebras also appear more naturally in measure theory than in probability theory.)

This motivates the definition of sigma algebras in measure theory: for a space $X$, some set of its subsets $\mathcal{F}\subseteq 2^X$ is called a $\sigma$-algebra if:

  • For sets $A_i\in\mathcal{F}$ ($i$ is countable), $\bigcup A_i\in\mathcal{F}$.
  • For set $A\in\mathcal{F}$, $A^c\in\mathcal{F}$.
And we define a measure $\mu:\mathcal{F}\to[0,\infty]$ on a sigma algebra as a map satisfying:
  • For disjoint $A_i$ ($i$ countable), $\mu\left(\bigcup A_i\right) = \sum\mu(A_i)$.
The tuple $(X,\mathcal{F})$ is called a measurable space and the tuple $(X,\mathcal{F},\mu)$ is called a measure space. The elements of the sigma algebra $\mathcal{F}$ are called measurable sets, and the basis of the sigma algebra is hopefully something simple and intuitive, like open intervals.

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