# Using the Denseness of $\mathbb{Q}$ for measurable functions

Consider two real-valued functions, $$f$$ and $$g$$ on $$\mathbb{R}$$.
Let L be, for a given $$a\in\mathbb{R}$$, $$L=\{x\in\mathbb{R}:f(x)+g(x)>a\}.$$

Why is it that we can re-write L equivalently as the following:

$$\bigcup_{r\in\mathbb{Q}}\{x\in\mathbb{R}:f(x)>a-r\}\cap\{x\in\mathbb{R}:g(x)>r\}.$$

It is stated it uses the fact that $$\mathbb{Q}$$ is dense, but I don't quite understand this equivalence.

These are discussed to show $$f+g$$ is measurable when $$f$$ and $$g$$ are each measurable in Stein and Shakarchi (2009).

Reference: $$\textit{Real Analysis: Measure Theory, Integration, and Hilbert Spaces}$$. Elias M. Stein, Rami Shakarchi. Princeton University Press, 2009.

Let $$x \in S=:\bigcup_{r\in\mathbb{Q}}\{x\in\mathbb{R}:f(x)>a-r\}\cap\{x\in\mathbb{R}:g(x)>r\}.$$

Then exists $$q \in \Bbb{Q}$$ such that $$f(x)>a-q$$ and $$g(x)>q$$ thus $$f(x)+g(x)>a-q+q=a$$

so $$x \in L:=\{x:f(x)+g(x)>a\}$$. Thus $$S \subset L$$

Now let $$x \in L$$ then $$f(x)+g(x)>a \Longrightarrow g(x)>a-f(x)$$.

By density of rationals exists $$q_0 \in \Bbb{Q}$$ such that $$g(x)>q_0>a-f(x)$$ thus $$f(x)>a-q_0$$ and $$g(x)>q_0$$ so $$x \in \{x\in\mathbb{R}:f(x)>a-q_0\}\cap\{x\in\mathbb{R}:g(x)>q_0\} \subset S$$

Thus $$L \subset S$$

The author chooses the density of the countable set of rationals to express the set $$L$$ as a countable union of measurable sets exploiting the measurability of $$f,g$$

• Brilliant. Thank you. Commented Sep 19, 2019 at 18:05
• The challenge for me was how did analysts come up with the set $S$? Commented Sep 19, 2019 at 18:08
• It is the idea i mention in the post...the denseness and countability of the rationals..you could not have a conclusion if you work with irrationals for instance. Commented Sep 19, 2019 at 18:11
• The denseness of $\mathbb{Q}$ is that you can always find another rational in the $\varepsilon$-nbhd of a rational, crudely put? Commented Sep 19, 2019 at 18:13
• Yes..in a more intuitive way, you can find a rational number between every two real numbers... Commented Sep 19, 2019 at 18:15