Countable indicator partition of a euclidean real func...

Countable indicator partition of a euclidean real function

Formulation 0

Let $f : X \to \mathbb{R}^D$ be a D4363: Euclidean real function such that

(i)	$E_0, E_1, E_2, \ldots \subseteq X$ are each a D78: Subset of $X$
(ii)	$E_0, E_1, E_2, \ldots$ is a D5143: Set partition of $X$

Then \begin{equation} f = \sum_{n = 0}^{\infty} f I_{E_n} \end{equation}

Subresults

Subresult 0
Countable indicator partition of a complex function

Let $f : X \to \mathbb{C}$ be a D4881: Complex function such that

(i)	$E_0, E_1, E_2, \ldots \subseteq X$ are each a D78: Subset of $X$
(ii)	$E_0, E_1, E_2, \ldots$ is a D5143: Set partition of $X$

Then \begin{equation} f = \sum_{n = 0}^{\infty} f I_{E_n} \end{equation}

Subresult 1
Countable indicator partition of a random euclidean real number

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$X : \Omega \to \mathbb{R}^D$ is a D4383: Random euclidean real number on $P$
(ii)	$E_0, E_1, E_2, \ldots \in \mathcal{F}$ are each an D1716: Event in $P$
(iii)	$E_0, E_1, E_2, \ldots$ is a D5143: Set partition of $\Omega$

Then \begin{equation} X = \sum_{n = 0}^{\infty} X I_{E_n} \end{equation}

Proofs