Lebesgue points for vector valued functions

Question

Let $Q$ be an open set in $\mathbb{R}^M$. Consider $ f \in L^1(Q;X)$, where $X$ is a Banach space. How does one show that the set of all Lebesgue points in $Q$ is of full measure? My idea was to use density of continuous functions but I'm not sure if continuous functions are dense in $L^1$ for any arbitrary open set $Q$ and Banach space $X$. Are continuous functions dense in the space $L^1(Q;X)$