Let $(X,L)$ be a compact polarized complex manifold of dimension $n$. Let $\varphi$ be a smooth positive metric on $L$. Define $\omega=dd^c\varphi$. We shall use $MA(\varphi)=\omega^n$ as the measure on $X$. Then there is a natural $L^2$-inner product on $H^0(X,L)$. Now fix $s\in H^0(X,L)$ of norm $1$. Consider the following set 
$$
A\_\epsilon=\{s'\in H^0(X,L): \|s'\|=1, |(s,s')|<\epsilon\},
$$
where $\epsilon>0$. My question is: how can we get an upper bound of
$$
MA(\varphi)\left(\bigcup\_{s'\in A\_{\epsilon}} Z\_{s'}\right),
$$
where $Z$ denotes the zero locus.



Bound of the measure of the support of a set of divisors in a fixed linear system
If I am not mistaken, it seems the set you are asking about is almost always $X$ itself (and the argument does not use anything about complex geometry). 


For any $p$ in $X$, the vector subspace


$$A\_p := \{ \sigma \in H^0(X,L) : \sigma(p) = 0 \}$$


has codimension at most 1 in $H^0(X,L)$.


On the other hand, for any $\epsilon$ your set $A\_\epsilon$ contains the subset


$$A\_0=\{s'\in H^0(X,L): \|s'\|=1, \ (s,s')=0\}.$$


Consider now the larger set 


$$B\_0=(s)^\perp = \{s'\in H^0(X,L): \ (s,s')=0\}.$$


Every element of $B\_0\setminus \{0\}$ is a nonzero multiple of an element of $A\_0$, so we have


$$\bigcup\_{s'\in B\_0 \setminus \{0\}} Z\_{s'} = \bigcup\_{s'\in A\_0} Z\_{s'} \subseteq \bigcup\_{s'\in A\_{\epsilon}} Z\_{s'}.$$


But now $B\_0$ is a vector subspace of $H^0(X,L)$ of codimension 1. So as long as $H^0(X,L)$ has dimension at least 3, for any $p \in X$ the intersection $A\_p \cap (B\_0 \setminus \{0\})$ will be nonempty, and so $p$ will be contained in $\bigcup\_{s'\in B\_0 \setminus \{0\}} Z\_{s'}$ and hence in the set you are asking about. 


