Positive Part of Simple Function is Simple Function
Jump to navigation
Jump to search
Theorem
Let $\struct {X, \Sigma}$ be a measurable space.
Let $f: X \to \R$ be a simple function.
Then $f^+: X \to \R$, the positive part of $f$, is also a simple function.
Proof
Let $f$ have the following standard representation:
- $f = \ds \sum_{i \mathop = 0}^n a_i \chi_{E_i}$
Then we see that $f^+$ must satisfy:
- $f^+ = \ds \sum_{i \mathop = 0}^n \max \set {a_i, 0} \chi_{E_i}$
as the $E_i$ are disjoint, and $\chi_{E_i} \ge 0$ pointwise.
Since all of the $E_i$ are measurable, it follows that $f^+$ is a simple function.
$\blacksquare$
Also see
- Negative Part of Simple Function is Simple Function, a natural counterpart
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $8.7 \ \text{(v)}$