Is there a special name for a measure over a power set?

Question

I have a set $\mathcal{L}$ and I want to study measures over $\mathcal{P}(\mathcal{L})$ or $2^\mathcal{L}$.

Answer 1

The power set is just the underlying set you're defining your measure for.

You could give any measure you come up with any name that you like.

Answer 2

I never came across a special name for a measure specifically defined over a power set.

Further I don't really think this would be necessary since all you need for a measure $\mu$ is essentially a $\sigma$-algebra of some kind - which the power set most certainly always is - I would always make it clear in context (if important) what kind of $\sigma$-algebra I am using.

The closest thing I can think of - where the $\sigma$-algebra as a power set is part of the definition - is the outer measure (wiki). So you could call your measure $\mu$ a $\sigma-additive$ outer measure.