Is $f^+$ a measurable function

Question

Let $f:X\to[-\infty,\infty]$ be a measurable function where $(X,\Sigma_x,\mu)$ is the measurable space.

Define $$ f^+(x)= \begin{cases} f(x) & \text{if }f(x)\ge0\\ 0 & \text{otherwise} \end{cases} $$

Is this a measurable function? if so, why?

Consider the inverse image of an open ray under the $f^+$. – 2017-02-18 — 2017-02-18

user id:130298 7k · Accepted Answer

Let $A=\{x:f(x)\ge 0\}$. Then $$f_+=f\cdot\chi_A$$ and $A$ is a measurable set. The product of measurable functions is measurable.

user id:154545 40k · Accepted Answer

Hint

$$f^+(x)=\boldsymbol 1_{f(x)\geq 0}(x)=\boldsymbol 1_{\{x\mid f(x)\geq 0\}}(x).$$

You want to multiply that indicator function by $f(x)$. – 2017-02-18 — 2017-02-18

user id:189130 14k · Accepted Answer

$f^+$ is the composition $p\circ f$ of $f$ and the continuous (hence Borel measurable) function $p:t\to t^+$.

3 Answers 3