Let $X\subseteq\mathbb{R}^n$ be a compact set and let $(X, \Sigma, \mu)$ be a measure space. Let $G$ be the space of all measurable functions $f:X\to R$ such that
(i) $\int|f(x)|^pd\mu(x)<\infty$ for some $p$ with $1\leq p<\infty$, and
(ii) $f$ satisfies a specific property $D$. Here, property $D$ can be monotonicity, convexity, supermodularity, etc. I am thinking $D$ is a property that uses the product order over $X$ and the standard order over $\mathbb{R}$ .
Is there a general theorem that says $G_C$, the space of all continuous measurable functions that satisfy (i)-(ii), is dense in $G$ (with the $L_p$-norm)? Or any suggestions how I should go about proving/disproving it?