Measure Space Definition Question

• December 13th 2009, 08:43 AM
Swlabr
Measure Space Definition Question
When we define a measure space $(X, \Sigma, \mu)$ do we take $\mu$ to be a (positive) measure ( $\Sigma$ the measurable sets)? That is, is $\mu$ a map $\mu: \Sigma \rightarrow [0, +\infty]$?

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?
• December 13th 2009, 09:38 AM
Moo
Hello,

$\Sigma$ will be the set ( $\sigma$-algebra) of all the $\mu$-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, $\mu$ would be a positive measure.