Show that if the function $f(x)$ is measurable on a set E, then the function $[f(x)]^a_b$ defined by $$[f(x)]^a_b:= \begin{cases} b, & \text{if }f(x)>b, \\ f(x), & \text{if }a\le f(x)\le b, \\ a, & \text{if }f(x)<a \end{cases}$$ is also measurable on E.
2026-03-30 20:56:00.1774904160
Is$[f(x)]^a_b:= \begin{cases} b, & \text{if }f(x)>b, \\ f(x), & \text{if }a\le f(x)\le b, \\ a, & \text{if }f(x)<a \end{cases}$measurable function?
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Let $A=\{x \in E: a \le f(x) \le b\}, B=\{x \in E: f(x)>b\}$ and $C=\{x \in E: f(x)<a\}$.
Since $f$ is measurable, the above sets are measurable.
We have
$[f(x)]^a_b=1_A f+1_B b+1_Ca$.
Can you procceed ?