Show that for any $1<p<\infty$ the set $\{ f \in L^p(\mathbb{R}) \cap L^1(\mathbb{R})\}$ where $ \int_{\mathbb{R}} f=0$ is dense in $L^p(\mathbb{R})$.

Question

Show that for any $1<p<\infty$ the set $\{ f \in L^p(\mathbb{R}) \cap L^1(\mathbb{R})\}$ where $ \int_{\mathbb{R}} f=0$ is dense in $L^p(\mathbb{R})$. Is the statement true if $\mathbb{R}$ is replaced by $[0,1]$? Also what can we say when $p=1$?

Any clues? Thanks

Answer 1

Lemma: Assume $p>1.$ Let $c\in \mathbb R.$ Then there exists a sequence $f_n\in L^1\cap L^p$ such that $\int f_n = c$ for all $n,$ and $\|f_n\|_p \to 0.$

Let's assume the lemma. Let $f\in L^1\cap L^p$ and put $c= - \int f.$ By the lemma, there are $f_n \in L^1\cap L^p$ such that $\int(f+f_n) = 0$ for all $n,$ with $\|f_n\|_p \to 0.$ Then $f+f_n \to f$ in $L^p,$ so we're done in the case $f\in L^1\cap L^p.$ Because $L^1\cap L^p$ is dense in $L^p,$ we've also proved the general result.

I'll leave the proof of the lemma to you for now.

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Second question: Here we work on $[0,1].$ The answer is no. Let $f\equiv 1.$ Suppose there are $f_n\in L^1\cap L^p$ with $\int f_n = 0$ such that $f_n \to f$ in $L^p.$ Because convergence in $L^p$ implies convergence in $L^1$ on sets of finite measure (by Holder), we then have $0 = \int f_n \to \int f =1,$ contradiction.