What is the 'largest' space of integrable functions which is also a Hilbert space?

Question

It is well known that $L^2(X,\mu)$, the set of functions $f:X \rightarrow \mathbb{C}$ such that $\int_X |f|^2 \text{d} \mu < \infty$, is a Hilbert space. Is there a Hilbert space $H$ such that $L^2(X,\mu)\subset H$? In other words, are there larger classes of integrable functions which form a hilbert space?

Answer 1

No. Let $V$ be the orthogonal complement of $L^2(X,\mu)$ in $H$ and let $f$ be a function in $V$.

Then, $\int fg d \mu =0$ for all $g$ in $L^2(X,\mu)$. This condition easily implies that $f$ is identically zero.