ACIT4330-Page/content/Definitions/Measure Theory/Simple Function.md

A simple function on X is a function s : X \to \mathbb{R} of the form s = \Sigma_{i=1}^{n} a_{i} \times X_{a_{i}} for pairwise disjoint A_{i} \subset X and distinct real numbers a_{i}

X_{A}(x) = \begin{cases}1 & \text{If}\ x \in A\\ 0 & \text{If}\ x \notin A\end{cases}

Note

If X has M then: s is Measurable \iff all A_{i} are Measurable

A_{i} = s^{-1}(\{ a_{i} \}) = s^{-1}(\mathbb{R} \setminus \{ a_{i} \})^{\complement}

If we have a Measure \mu on X, define \int_{A} s \, d\mu \equiv \Sigma_{i=1}^{n} a_{i} \times \mu (A \cap A_{i}) for A \in M (they are all \in [0, \infty]).