Measure spaces and measurable sets

Ina e-mailed that she was trying to understand what it means for a set to be measurable. Here was my quick writeup:

As the Wikipedia entry for Measure Theory says, "a measure is a function that assigns a number, e.g., a 'size', 'volume', or 'probability', to subsets of a given set."

Using the notation of that Wikipedia entry, which is pretty standard, the "given set" is X, the subsets are the members of a σ-algebra Σ over X, and the measure is a function μ: Σ -> [0, ∞]. So: a measure is a function μ on a σ-algebra Σ over a set X. A measure space consists of those 3 things: it's a triple (X, σ, μ).

So μ maps every Y \in Σ to a real number in [0, ∞], and μ(Y) is the "size" of Y under that measure. Then the measurable sets are just the sets in the σ-algebra Σ: all the sets that "measures."

Maybe I should contribute some of this to the Wikipedia entry...