How prove that, $C(\mathbb{R})\cap L^p(\mathbb{R})$ is dense in $L^p(\mathbb{R})$ ? Thanks
$C(\mathbb{R})\cap L^p(\mathbb{R})$ is dense in $L^p(\mathbb{R})$
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real-analysis
lp-spaces
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0What construction of $L^p$ do you use? What have you tried? Where do you get stuck? – 2017-01-16
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0See Donald Cohn, Measure Theory, proposition 7.4.3, page 207: even the compactly supported functions are dense in $L^p(X)$, if $p<\infty$ (and $X$ is a locally compact Hausdorff space, $A$ is a $σ$-algebra on $X$ that includes the Borel sets, and $\mu$ is a regular measure on $(X,A)$). – 2017-01-16
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0I think that you should have $p<\infty$... – 2017-01-16
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We have that $C_c(\mathbb{R})$ are dense in $L^p(\mathbb{R})$ and $C_c(\mathbb{R})\subset C(\mathbb{R})∩L^p(\mathbb{R}) \subset L^p(\mathbb{R})$.
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0I think that you should have $p<\infty$... Moreover, why is $C_c$ dense in $L^p$? Your answer should be a comment, in my opinion (except if the OP already knows that $C_c$ dense in $L^p$). – 2017-01-16
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0Yes ,we must have $1\leq p<\infty$. – 2017-01-16