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Show a limited function is measurable

  • #1
Not sure about the translated term limited (from German); perhaps cut-off function?

Homework Statement



Let [itex]f[/itex] be a measurable function in a measure space [itex](\Omega, \mathcal{F}, \mu)[/itex] and [itex]C>0[/itex]. Show that the following function is measurable:
[tex]f_C(x) =
\left\{
\begin{array}{ll}
f(x) & \mbox{if } |f(x)| \leq C \\
C & \mbox{if } f(x) > C \\
-C & \mbox{if } f(x) < -C
\end{array}
\right.[/tex]

Homework Equations



None in particular. Definition, measurable space:
An ordered tuple [itex](\Omega,\mathcal{F})[/itex], where [itex]\Omega[/itex] is a set and [itex]\mathcal{F}[/itex] is a [itex]\sigma[/itex]-algebra of subsets in [itex]\Omega[/itex], is called a *measurable space*.

Definition, measureable function:
https://en.wikipedia.org/wiki/Measurable_function

The Attempt at a Solution



Since [itex]f[/itex] is measurable, then [itex]f_C[/itex] is measurable when [itex]|f(x)| < C[/itex].

It should be trivial to prove that a constant function is measurable.

I'm not sure how to approach [itex]f_C[/itex] at [itex]C[/itex]. Perhaps: We have shown that [itex]f_C[/itex] is measurable at all points except [itex]f(x)=C[/itex], but a single point has measure [itex]0[/itex]. However this seems very hand-wavy and probably entirely incorrect...
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618
You can make it a lot less hand-wavy if you state and work with the definition of 'measurable'. You want to show ##\{x:f_C(x)<a\}## is measurable for all ##a##. You know ##\{x:f(x)<a\}## is measurable for all ##a##. How do they compare?

Not sure about the translated term limited (from German); perhaps cut-off function?

Homework Statement



Let [itex]f[/itex] be a measurable function in a measure space [itex](\Omega, \mathcal{F}, \mu)[/itex] and [itex]C>0[/itex]. Show that the following function is measurable:
[tex]f_C(x) =
\left\{
\begin{array}{ll}
f(x) & \mbox{if } |f(x)| \leq C \\
C & \mbox{if } f(x) > C \\
-C & \mbox{if } f(x) < -C
\end{array}
\right.[/tex]

Homework Equations



None in particular. Definition, measurable space:
An ordered tuple [itex](\Omega,\mathcal{F})[/itex], where [itex]\Omega[/itex] is a set and [itex]\mathcal{F}[/itex] is a [itex]\sigma[/itex]-algebra of subsets in [itex]\Omega[/itex], is called a *measurable space*.

Definition, measureable function:
https://en.wikipedia.org/wiki/Measurable_function

The Attempt at a Solution



Since [itex]f[/itex] is measurable, then [itex]f_C[/itex] is measurable when [itex]|f(x)| < C[/itex].

It should be trivial to prove that a constant function is measurable.

I'm not sure how to approach [itex]f_C[/itex] at [itex]C[/itex]. Perhaps: We have shown that [itex]f_C[/itex] is measurable at all points except [itex]f(x)=C[/itex], but a single point has measure [itex]0[/itex]. However this seems very hand-wavy and probably entirely incorrect...
 

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