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Section 2.1 Integrable Sets

Subsection 2.1.1 Characteristic Functions

In single variable calculus, integrating over intervals is sufficient for basically any purpose. Even a more complicated subset of \(\RR\) for integration is almost certainly something that can be broken down into a number of intervals.

For multivariable integrals, the situation is very different. The domain of a scalar field can be any kind of strange subset of \(\RR^n\text{,}\) not necessarily the nice rectangles or rectangular prisms that are intervals. We may want to integrate over circular domains, or any kind of more complicated shapes. This section builds up the theory to understand such integrals: how to move from integration over intervals to integration over arbitrary domain in \(\RR^n\text{.}\)

First, not all domains are possible. I said ‘arbitrary’ domains, but there are some restrictions. Some formal definitions are needed here.

Definition 2.1.1.

Let \(S\) be a subset of \(\RR^n\text{.}\) The characteristic function of \(S\) is defined as follows, where \(v \in \RR^n\text{.}\)

\begin{equation*} \chi_S (v) : = \left\{ \begin{matrix} 1 \amp v \in S \\ 0 \amp v \notin S \end{matrix} \right. \end{equation*}

The characteristic function is very simple: it has value \(1\) on the set \(S\) and \(0\) everywhere else.

Subsection 2.1.2 Integrable Sets

The characteristic function lets us make another formal definition.

Definition 2.1.2.

Let \(S\) be a subset of \(\RR^n\) which is contained in an interval \(I\text{.}\) \(S\) is called a integrable set if the following integral exists.

\begin{equation*} \int_I \chi_S dV \end{equation*}

Almost all sets we will work with are integrable. Sets which are not integrable are very strange sets (at least, for doing anything geometric). The archetypical example is \(\QQ \subset \RR\text{:}\) \(\QQ\) is not an integrable subset of \(\RR\text{.}\) What does is mean, for a single variable function, to integrate over all rational points but not over the irrational points? Since the rational and irrational points are everywhere in \(\RR\text{,}\) there is no notion of ‘area under a curve’ here at all. This integral doesn't make any sense; the set it not an integrable set. Equivalently, the Riemann integral limit doesn't not converges, since you can always choose rational or irrational \(x_k^*\text{;}\) the different choices lead to different values of the limit for these choices, so the integral doesn't work.

As I said, this is a formal definition. It won't be necessary to check integrability of the domain each time I do an integral. All reasonably geometric sets will be integrable. But the formal definition of an integrable set is necessary to give the the next formal definition. As with most formal definitions this is pretty useless for actual calculations. We'll move on to calculations in Section 2.2 after we make the proper, formal definitions.

Subsection 2.1.3 Integration over Integrable Sets

Formally, let me now define how to integrate an integrable function over any integrable set.

Definition 2.1.3.

Let \(S\) be a integrable subset of \(\RR^n\) with \(S \subset I\) for some interval \(I\text{.}\) Let \(f: S \rightarrow \RR\) be an integrable function which can be extended to an integrable function \(\tilde{f}\) on \(I\text{.}\) Then the integral of \(f\) over \(S\) is defined as:

\begin{equation*} \int_S f dV \defeq \int_I \tilde{f} \chi_S dV \end{equation*}

The characteristic function removes all values of \(f\) outside the set \(S\text{,}\) so all that the integral measure are the values of \(f\) over \(S\text{.}\)

I mentioned earlier that all open sets are integrable. I have another simplifying theorem that ensure we only need to integrate over open sets.

This proposition means that we can ignore any inconveniences which happen on the boundaries of sets. (Note that we still do need limits for improper integrals, when the value of a function approaches \(\infty\) near the boundary of a set). The proposition also means that sets which are essentially of lower dimensions do not matter in integration. If we are integrating over subsets of \(\RR^3\text{,}\) then integrating over points, line and planes amounts to nothing. Points, lines, and planes don't have any interior points in \(\RR^3\text{,}\) so we might as well be integrating over the empty set, which gives zero.

Other than ignoring boundaries and smaller dimensional pieces, our approach to general sets is to break them up into pieces. It is a general result in topology, which we won't get into here, that any (reasonable) set can be broken up in to pieces (possibly infinitely many) where the previous style of integral applies. We can also be creative: for example, we could integrate over a larger circle and subtract the integral over a small circle to integrate over a ring-shaped domain. In these various ways we can set up integrals over any (reasonable) subset of \(\RR^n\text{.}\)