Real Analysis, Folland Corollary 2.9

This follows from propostion 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.

Corollary 2.9: If $\{f_j\}$ is a sequence of complex-valued measurable functions and $f(x) = \lim_{j\rightarrow \infty}f_j(x)$ exists for all $x$, then $f$ is measurable.

Proof: We have $\{f_j\}_{j\in \mathbb{C}}$. From Corollary 2.5 we know that $\mathcal{B}_{\mathbb{C}} = \mathcal{B}_{\mathbb{R}^2} = \mathcal{B}_{\mathbb{R}}\otimes \mathcal{B}_{\mathbb{R}}$, and we are given the fact that $f(x) = \lim_{j\rightarrow \infty}f_j(x)$ exists for all $x$, therefore $f: X \rightarrow \mathbb{C}$, and again from Corollary 2.5, $f$ is measurable.

I just want to verify that this is indeed correct, if not any suggestions or hints is greatly appreciated.

Corollary 2.9: If $\{f_j\}$ is a sequence of complex-valued measurable functions and $f(x) = \lim_{j\rightarrow \infty}f_j(x)$ exists for all $x$, then $f$ is measurable.

Let us proof it step by step.

Step 1. As a preliminary step, let us prove a corollary of Proposition 2.7.

If $\{f_j\}$ is a sequence of $\mathbb{R}$-valued measurable functions on $(X,M)$ and $f(x) = \lim_{j\rightarrow \infty}f_j(x)$ exists for all $x$, then $f$ is measurable.

Proof: In fact, if $\{f_j\}$ is a sequence of $\mathbb{R}$-valued measurable functions on $(X,M)$ and $f(x) = \lim_{j\rightarrow \infty}f_j(x)$ exists for all $x$, then $\{f_j\}$ is a sequence of $\overline{\mathbb{R}}$-valued measurable functions on $(X,M)$, and since $f(x) = \lim_{j\rightarrow \infty}f_j(x)=\limsup_{j\rightarrow \infty} f_j(x)$, we have that $f$ is measurable.

Remark: if you prefer $\liminf$ you could use $f(x) = \lim_{j\rightarrow \infty}f_j(x)=\liminf_{j\rightarrow \infty} f_j(x)$, to conclude that $f$ is measurable.

Step 2. Now let us prove Corollary 2.9.

Suppose $\{f_j\}$ is a sequence of complex-valued measurable functions and $f(x) = \lim_{j\rightarrow \infty}f_j(x)$ exists for all $x$.

Then, using Corollary 2.5, we have that $\textrm{Re}(\{f_j\})$ is measurable, for all $j$. So, we have that $\textrm{Re}(\{f_j\})$ is a sequence of $\mathbb{R}$-valued measurable functions on $(X,M)$ and $\textrm{Re}(f(x)) = \lim_{j\rightarrow \infty}\textrm{Re}(f_j(x))$, for all $x$. So by step 1, we have that $\textrm{Re}(\{f\})$ is measurable.

In a similar way, using Corollary 2.5, we have that $\textrm{Im}(\{f_j\})$ is measurable, for all $j$. So, we have that $\textrm{Im}(\{f_j\})$ is a sequence of $\mathbb{R}$-valued measurable functions on $(X,M)$ and $\textrm{Im}(f(x)) = \lim_{j\rightarrow \infty}\textrm{Im}(f_j(x))$, for all $x$. So by step 1 again, we have that $\textrm{Im}(\{f\})$ is measurable.

So we concluded that $\textrm{Re}(\{f\})$ and $\textrm{Im}(\{f\})$ are measurable. Then, from Corollary 2.5, we have that $f$ is measurable.

Honestly I'm not entirely sure what you're saying.

$f$ is from $X$ to $\Bbb{C}$. By Corollary 2.5 it suffices to show that the real and imaginary parts are measurable as you observed, but the $\mathrm{Re}$ and $\mathrm{Im}$ functions are continuous, so

$$\mathrm{Re}\, f = \lim_{j\to\infty} \mathrm{Re}\, f_j.$$ and similarly for the imaginary part. Hence it suffices to show that the pointwise limit of a sequence of measurable real valued functions is measurable when it exists (as it does in this case). Try to prove this from proposition 2.7 (thought it seems like you may have already, like I said, I find your question confusing.).

Solution:

When the limit exists, it is equal to the lim sup. Since the lim sup is measurable, so is the limit.

• I wasn't sure either I made the post awhile back and I still don't really know what to do. Folland simply said apply Corollary 2.5. – Wolfy Jun 21 '16 at 4:37
• Ok, well I explained basically how to do it. Does it make sense? Feel free to ask a question if what I wrote is unclear. I'll do my best to clarify. I don't always explain entirely clearly, despite my best efforts. – jgon Jun 21 '16 at 4:38
• I am not really sure why the real and imaginary parts of $f$ are continuous, (I have not been able to prove corollary 2.5). As for proposition 2.7 I was only able to prove the first part i.e., that $g_1,g_2,g_3,g_4$ are measurable. – Wolfy Jun 21 '16 at 4:45
• Correct me if I am wrong but it seems that the real and imaginary parts of $f$ are continuous by Corollary 2.2. If not then, there are measurable functions that are not continuous? – Wolfy Jun 21 '16 at 4:49
• Ah, it's not that the real and imaginary parts of $f$ are continuous, but that taking the real part of a complex number is continuous. – jgon Jun 21 '16 at 11:33