Let $(\Omega,\mu)$ be a measure space, and let $X=L^{2}(\Omega,\mu)$ be the complex Hilbert space of square-integrable complex measurable functions on $\Omega$. (Each $f \in L^{2}$ is an equivalence class of measurable functions which are equal a.e. $[d\mu]$.)
If $f$ is a measurable function on $\Omega$, define $M_{f}$ to be the linear subspace of $X$ consisting of all $g \in L^{2}(\Omega,\mu)$ such that $fg\in L^{2}(\Omega,\mu)$.
Prove or disprove: $M_{f}$ is dense in $X$ for every real measurable $f$.
Hint 1: By replacing $f$ with a related function, show that it suffices to consider $f \ge 1$.
Hint 2: Try to show $M_f^\perp = 0$.
Hint 3: Suppose $h \in M_f^\perp$; try to show $\|h\|^2 = 0$. Think about multiplying by $f/f$.