Question 13

Question

What is a measure? Easy examples?

Answer

It is pretty much self explanatory with the name for what it is supposed to do.

But a measure on a set $X$ (maybe not required: with $\Sigma$ as a sigma-algebra over $X$)is a function $\mu :M\to [0,\infty ]$, such that:

1. Non-negativity: for all $E\in \Sigma ,\mu (E)\ge 0$. (maybe not required)
2. $\mu (\varnothing )=0$
3. Countable additivity: $\mu ({\cup }_{n=1}^{\infty }{A}_n)={\sum }_{n=1}^{\infty }\mu (A_n)$ (for pairwise disjoint $A_n\in M$)

Then we say that $X$ is a measure space.

Note

You may also want to take a look at measurable

Easy Examples