Continuous Functions with Compact Support Dense in Lebesgue P-Space

Theorem

Let $\map {C_c} {\R^n}$ be the space of continuous functions with compact support on $\R^n$.

Let $p \in \R$ such that $p \ge 1$.

Let $\map {\LL^p} {\lambda^n}$ be the Lebesgue $p$-space for Lebesgue measure $\lambda^n$.

Then $\map {C_c} {\R^n}$ is everywhere dense in $\map {\LL^p} {\lambda^n}$ with respect to the $p$-seminorm $\norm {\, \cdot \,}_p$.

Proof

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Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $14.7$