Measure space

Before we can define Measure space we need a measurable space.

Measurable Space

Let $X$ be a set and $\mathcal{A}$ a $\sigma$-algebra, then $(X,\mathcal{A})$ is a Measurable space

Measure space

A measurable space and a function, $$\mu:\mathcal{A}\rightarrow[0,\infty]$$ which is a measure is a measure space, that is a measure space is:

$$(X,\mathcal{A},\mu:\mathcal{A}\rightarrow[0,\infty])$$ but recall mathematicians are lazy so we just write $$(X,\mathcal{A},\mu)$$