Why is this property true of sets/indicator functions? (A page from my textbook)

[jdinatale]

I can't for the life of me figure out why the underlined sentence in red is true:

Could someone clarify?

 

[LCKurtz]

That page doesn't tell us what $E$ or $S_0(E)$ represent. Nor what the canonical representation of $\phi \in S_0(E)$ is.

 

[jdinatale]

That page doesn't tell us what $E$ or $S_0(E)$ represent. Nor what the canonical representation of $\phi \in S_0(E)$ is.

My apologies,

$E$ is a measurable set.

$S_0(E)$ is the set of simple functions on $E$.

The canonical representation of $\phi \in S_0(E)$ is a linear combination of characteristic functions $\phi = \sum_{i=1}^na_i\mathcal{X}_{E_i}$.

 

[LCKurtz]

My apologies,

$E$ is a measurable set.

$S_0(E)$ is the set of simple functions on $E$.

The canonical representation of $\phi \in S_0(E)$ is a linear combination of characteristic functions $\phi = \sum_{i=1}^na_i\mathcal{X}_{E_i}$.

I'm guessing there is more to the definition of canonical representation. Maybe you are given that the $A_i$ are disjoint with each other and $B_j$ similarly. And if $\cup_i A_i = E$ and $\cup_j B_j = E$ that would explain those equations.