Let $(V, |\cdot|_V)$ be a Banach space, $X$ be a metric space, $(X, F, \mu)$ be a measure space such that for every compact $K\subseteq X$, $K \in F$ and $\mu (K) < \infty$, and $L_p(X, \mu, V)$ be a Banach space with norm $$\|f\|=\left(\int_X \big(\vert f\rvert_V\big)^p d\mu\right)^{1/p}.$$ Let $A \in F$ be a compact set and $(B_n)_{n\geq 1}$ be a sequence of measurable subsets of $A$ such that $\mu (B_n) \rightarrow 0$ when $n\rightarrow \infty$. For $B\in F$ we define $I_B f=\left(\int_B \big(\vert f\rvert_V\big)^p d\mu\right)^{1/p}$.
Does $I_{B_n} f \rightarrow 0$ when $n \rightarrow \infty$?
If not in general then are there some constraints on $X$, $\mu$ and $V$ that ensure that? Or does an additional assumption that $(B_n)_{n\geq 1}$ is a descending sequence with respect to $\subseteq$ help?
Edit:
I forgot to add that every compact set is in $F$. I edited it in now.
What we have to show is that if $g$ is an integrable function over a measure space and $\mu(B_n)\to 0$, then $\int_{B_n}|g|\mathrm d\mu\to 0$.
(in order to be reduced to this case, we have to assume that $\lVert f\rVert_V$ is Borel($V$)-Borel($\mathbb R$) measurable)
We can approximate $g$ in $\mathbb L^1(\mu)$ by linear combinations of characteristic functions of measurable sets, hence we only need to prove it when $g$ is a characteristic function of a measurable set. But this relies on the inequality $\mu(A\cap B_n)\leqslant \mu(B_n)$ valid for any $n$ and any measurable set $A$.