Support of a measure and set where it is concentrated I have some problems understanding the difference between the notion of support of a measure and saying that a measure is concentrated on a certain set. If I have a $\sigma$-algebra and $\mu$ a measure, is it equivalent to ask the support of $\mu $ to be contained in a measurable set $A$ and $\mu$ to be concentrated on $A$?

The support of a measure is the intersection of all closed sets that have full measure. This means that if a measure $\mu$ is concentrated on a set $A$, the inclusion $\operatorname{supp}(\mu) \subset \overline{A}$ holds [if $\mu$ is at least $\sigma$-finite].

This is pretty much all you can say about the set-theoretic relation between those two sets. Take for example the Lebesgue-measure. It's support is $\mathbb{R}$, but it is concentrated on every set of the form $\mathbb{R} \setminus \\{x\\}$. You can remove any element from $\mathbb{R}$ and the Lebesgue-measure is still concentrated on this "narrowed" set. But the support is uniquely determined as $\mathbb{R}$.