Result on D41: Indicator function
Subresult to R4565: Countable indicator partition of a euclidean real function
Countable indicator partition of a random euclidean real number
Formulation 0
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
Proof 0
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 $\Omega$
This result is a particular case of R4565: Countable indicator partition of a euclidean real function. $\square$