Measure Space Definition Question

When we define a measure space do we take to be a (positive) measure ( the measurable sets)? That is, is a map ?

Or am I just confused? Will it just depend on who is defining it? On the book I am reading? Or is it by convention?

Hello,

will be the set ( -algebra) of all the -measurable sets.

But then I don't know if talking about a measure space this way implies that the measure is positive... In my opinion, unless clearly stated, would be a positive measure.