Let $E \subset \Bbb R^d$ such that $m(E)=0$ and $f : E \longrightarrow \Bbb R$ is continuous. Is the space of all integrable continuous functions $F :\Bbb R^d \longrightarrow \Bbb R$ with $F|_E =f$ dense in $L^1 (\Bbb R^d)$?
How can I proceed to tackle this problem? Please help me.
Thank you very much.
The answer is no. For example, let $E=\mathbb Q$. Since any continuous function is determined by the values it takes on $\mathbb Q$, there will be at most one continuous extension of any $f:\mathbb Q\to \mathbb R$, so the set of such extensions is far from dense.