LEC CAL2 5

Integrating over Jordan measurable sets

Definition 196.1(Definition).

If $C\subset {\mathbb{R}}^n$, the characteristic function ${\chi }_C$ of $C$ is defined by

$${\chi }_C=\{\begin{array}{ll}0& x\notin C\\ 1& x\in C.\end{array}$$

The support of ${\chi }_C$ is given by $\overline{C}$.

Definition 196.2(Definition).

Assume $C\subset {\mathbb{R}}^n$ is bounded, that is, $C\subseteq R$ for some rectangle $R$. Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be bounded on $R$. $f$ is said to be integrable over $C$ if $f{\chi }_C$ is integrable over $R$, in which case we define

$${\int }_{C}f:={\int }_{R}f{\chi }_C.$$

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Check that $R$ can be replaced by any rectangle $R^{\prime }$ which also contains $C$ and on which $f$ is bounded.

$f$ and ${\chi }_C$ being integrable is a simple and often used sufficient condition for $f{\chi }_C$ being integrable.

The boundary $\partial C$ of a set $C\subseteq {\mathbb{R}}^n$ is defined to be $\overline{C}\setminus C^{\circ }$.

Theorem 196.3(Theorem).

Let $C\subseteq R$ as above. Then ${\chi }_C:R\to \mathbb{R}$ is integrable on $R$ $⟺$ the boundary of $C$ has measure zero.

Proof.

It suffices to show that$\partial C$ is the set of discontinuities of ${\chi }_C$. This is simple enough: $x\in \partial C$ $⟺$ $x\notin C^{\circ }$ and $x$ is a limit point of $C$ $⟺$ every neighborhood of $x$ contains a point of $C$ and a point of $C^c$ $⟺$ ${\chi }_C$ is discontinuous at $x$.□

Definition 196.4(Definition).

A bounded set is called Jordan measurable if its boundary has measure zero.

Note that the boundaries of Jordan measurable sets also have content zero since they are compact.

The integral ${\int }_{C}1={\int }_R{\chi }_C$ is called the ($n$-dimensional) content/volume of $C$.