When we define a measure space $\displaystyle (X, \Sigma, \mu)$ do we take $\displaystyle \mu$ to be a (positive) measure ($\displaystyle \Sigma$ the measurable sets)? That is, is $\displaystyle \mu$ a map $\displaystyle \mu: \Sigma \rightarrow [0, +\infty]$?

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?