Let $1\le p <\infty$ and let $(X,\mathscr{M},\mu)$ be a measure space. Let $$S:=\{ f:X\to \mathbb{C}:f\text{ measurable, }f(X)\text{ is a finite set, and }\mu(\{ x \in X:f(x)\not= 0 \})<\infty\}$$then $S$ is a Dense subspace in $L^{p}(X,\mathscr{M},\mu)$ .