Integral of Compactly Supported Function

Theorem

Let $f : \R \to \R$ be a continuous real function.

Let $K \subset \R$ be a compact subset, say, $\closedint a b$.

Let $K$ be the support of $f$:

$\map \supp f = K$.

Then:

$\ds \int_{- \infty}^\infty \map f x \rd x = \int_a^b \map f x \rd x$

Proof

\(\ds \int_{-\infty}^\infty \map f x \rd x\)

\(=\)

\(\ds \int_{\overline \R} \map f x \rd x\)

Definition of Extended Real Number Line

\(\ds \)

\(=\)

\(\ds \int_{K \cup \paren { {\overline \R} \setminus K} } \map f x \rd x\)

Definition of Set Difference, Definition of Set Complement

\(\ds \)

\(=\)

\(\ds \int_K \map f x \rd x + \int_{\R \setminus K} \map f x \rd x\)

Riemann Integral Operator is Linear Mapping

\(\ds \)

\(=\)

\(\ds \int_K \map f x \rd x\)

Definition of Support of Continuous Mapping

\(\ds \)

\(=\)

\(\ds \int_a^b \map f x \rd x\)

$\blacksquare$