# Real Analysis, Folland Corollary 2.8

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.

• Take $f_k =f$ for odd $k$ and $f_k = g$ for even $k$, then $\max(f,g) = g_1, \min(f,g) = g_2$. – copper.hat Oct 29 '15 at 18:09
• @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. – David C. Ullrich Oct 29 '15 at 18:19
• @DavidC.Ullrich: True, I didn't even think about the implications... – copper.hat Oct 29 '15 at 18:24
• 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? – Wolfy Oct 29 '15 at 18:29
• 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). – Nate Eldredge Oct 29 '15 at 19:08

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.