1
$\begingroup$

This Corollary follows from proposition 2.7:

If $\{f_j\}$ is a sequence of $\overline{\mathbb{R}}$-valued measurable functions on $(X,M)$, then the functions $$\begin{aligned} g_1(x) = \sup_{j}f_j(x), \ \ \ \ g_3(x) = \lim_{j\rightarrow \infty}\sup f_j(x) \end{aligned}$$ $$\begin{aligned} g_2(x) = \inf_{j}f_j(x), \ \ \ \ g_4(x) = \lim_{j\rightarrow \infty}\inf f_j(x) \end{aligned}$$ are all measurable functions. If $f(x) = \lim_{j\rightarrow \infty}f(x)$ exists for every $x\in X$, then $f$ is measurable.

If you want a proof for this let me know I will post it.

Corollary 2.8: If $f,g:X\rightarrow \overline{\mathbb{R}}$ are measurable, the so are $\max(f,g)$ and $\min(f,g)$

Proof (suggested from CopperHat user): Suppose we have $f,g:X\rightarrow \overline{\mathbb{R}}$ are measurable, define $f_k = f$ for odd $k$ and $f_k = g$ for even $k$. So, if $f < g \Rightarrow \sup_k f_k(x) = f$ and if $f > g \Rightarrow \sup_k f_k(x) = g$. So then, $\max(f,g) = g_1$. The same argument can be applied for $\min(f,g)$, hence $\max(f,g)$ and $\min(f,g)$ are measurable.

$\endgroup$
13
  • 1
    $\begingroup$ Take $f_k =f $ for odd $k$ and $f_k = g$ for even $k$, then $\max(f,g) = g_1, \min(f,g) = g_2$. $\endgroup$ – copper.hat Oct 29 '15 at 18:09
  • $\begingroup$ @copper.hat Indeed - so 2.8 follows from 2.7. But I can't see how it's possible for a person to be able to prove 2.7 without being able to prove 2.8... very curious. $\endgroup$ – David C. Ullrich Oct 29 '15 at 18:19
  • $\begingroup$ @DavidC.Ullrich: True, I didn't even think about the implications... $\endgroup$ – copper.hat Oct 29 '15 at 18:24
  • $\begingroup$ Well, I use proposition 2.3 and the definition of limsup and liminf to prove proposition 2.7, are you suggesting I go about the same approach to prove the corollary? $\endgroup$ – Wolfy Oct 29 '15 at 18:29
  • $\begingroup$ Alternatively, take $f_1 = f$, $f_2 = g$, and $f_3 = f_4 = \dots = -\infty$, so then $\max(f,g) = \sup f_n$. I think maybe "Corollary" should have instead said "Useful Special Case". In fact rather than deducing 2.8 as a corollary of 2.7, it might be better to simply inspect the proof of 2.7 and see that the same idea proves 2.8 (presumably you will replace a countable union somewhere with a union of two sets). $\endgroup$ – Nate Eldredge Oct 29 '15 at 19:08
1
$\begingroup$

Let $f_k = f$ for $k$ odd and $f_k = g$ for $k$ even.

Then $\max(f(x),g(x)) = \max(f(x),g(x),f(x),g(x),...) = \sup_k f_k(x) = g_1(x)$.

Hence by Proposition 2.7 $g_1$ and hence $x \mapsto \max(f(x),g(x))$ is measurable.

Similarly, $g_2$ is measurable.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.