Are $L_p$ spaces of functions with separable support separable?

Question

Let $X$ be a separable space. Is $L_p$$(X, \mu, V)$ a separable space?

Here, $(V, |\cdot|_V)$ is a normed space. And a norm of $L_p(X, \mu, V)$ is: $$ \|f\|=\left(\int_X \big(\vert f\rvert_V\big)^p d\mu\right)^{1/p}. $$ If it's not a separable space, then what about the usual case of $V=\mathbb R$?

Edit:
As suggested in the comments let's assume $V$ is separable. Also we might consider a $\sigma$-finite measure $\mu$.

Edit 2:
Lastly what if $X$ would be a Polish space?

Answer 1

If $V$ is not separable, then $L^p(X,\mu,V)$ is not either.

Take $\{v_j\}$ an uncountable set in $V$ without a limit point and $f_j\in L^p(X,\mu,V)$, such that $f_j(x)=\varphi_j(x)v_j$, where $\varphi_j\ne 0$ scalar. Clearly, there is no limit point in $\{f_j\}$.

If $X$ is separable, then again it is not certain that $L^p(X,\mu,\mathbb R)$ is separable. It depends on the measure $\mu$. If $\mu$ is a counting measure, on the Borel sets, and $X$ uncountable, then clearly $L^p(X,\mu,\mathbb R)$ is not separable, as all the $\delta$-function (which are measurable) do not have any limit point.