Difference between “measure on” and “measure over”

Question

I want to make sure I understand the difference between the terms "measure on" and "measure over," assuming there is one. Is a measure on the set $X$ the same as a measure over its power set $\mathfrak{P}(X)$? To talk about a measure that is defined only for certain subsets of $X$, say for a $\sigma$-algebra $\mathcal{B} \subset \mathfrak{P}(X)$, or for the relativized power set $\mathfrak{P}(X)^M$ in an inner model $M$ containing $X$, would one say "measure over $\mathcal{B}$" and "measure over $M$" respectively? In the second case, would one say "measure on $X$ over $M$"?

Answer 1

From searching the web and some books in my shelf it seems that it usually depends on how you have worded rest of the phrase. Wikipedia for example uses "measure on" when referring to the underlying set and the $\sigma$-algebra, if no further specifications are made. Here it states "measure on the Borel algebra": http://en.wikipedia.org/wiki/Borel_set , and here it says "measure on $\mathbb{R}^{n}$": http://en.wikipedia.org/wiki/Lebesgue_measure .

My impression was that at least in the following two cases there might be a small difference.

1. If $\mu$ is e.g. a Borel-measure, one would say that "$\mu$ is a measure on the Borel $\sigma$-algebra", and "$\mu$ is a measure over the Borel sets". So when naming the $\sigma$-algebra alone one would use "measure on", and when specifying contents of the $\sigma$-algebra one uses "measure over". Here http://en.wikipedia.org/wiki/Sigma-algebra , for example, Wikipedia says "the collection of sets over which a measure is defined is a σ-algebra".

2. If $X$ is a set and $M$ a $\sigma$-algebra, one would say that "$\mu$ is a measure over $(X,M)$", and that "$\mu$ is a measure on $X$" or "$\mu$ is a measure on $M$". I.e. "over" when referring to the pair, and "on" when referring to the individual parts of the pair.

So in your case, I guess you could omit mentioning $X$ if it is known to be mutual for all the $\sigma$-algebras. Or you could say "a measure over $(X,M)$", for example, instead of "a measure on $X$ over $M$". But in any case, I don't think there is any serious harm made if you choose to use either of the expressions.

Hope this helps.