# Equivalence of definition of measure concentrated on a set $A$

Let $\lambda$ be a positive or complex measure. We say that $\lambda$ is concentrated on $A$ if for some set $A \in \mathcal{B }$ we have that $\lambda (E) =\lambda (A \cap E )$ for every $E \in \mathcal {B }$ , where $\mathcal {B }$ is some $\sigma$-algebra.

I want to show that this is equivalent to the hypothesis that $\lambda (E) =0$ whenever $E \cap A = \emptyset$.

Suppose $\lambda (E) =0$ whenever $A \cap E = \emptyset$. Now let $F$ be any set, then $F = (F \cap A) \cup ( F \cap A ^c )$, and since this is a disjoint union and $(F \cap A^c)\cap A = \emptyset$ so that $\lambda(F \cap A^c)=0$. We then have that $\lambda (F)=\lambda (F \cap A )$.
Conversely, suppose $\lambda (E)=\lambda(E \cap A)$, then if $E \cap A=\emptyset$, $\lambda(E)=\lambda(\emptyset)=0$.