ACIT4330-Page/content/Revision/Real Analysis/Question 13.md

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) \geq 0. (maybe not required)
2. \mu(\emptyset) = 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

Lecture 14#Lemma