Exercise 1.2.1 (Countable unions/intersections of Jordan measurable sets need not be Jordan measurable)

Question

Show that the countable union $\bigcup_{n=1}^{\infty} E_n$ or countable intersection $\bigcap_{n=1}^{\infty} E_n$ of Jordan measurable sets $E_1,E_2,\ldots \subseteq \mathbb{R}$ need not be Jordan measurable, even when bounded.
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Elu Thingol · Accepted Answer

Motivated by the \href{./exercise_1-1-18}{Exercise 1.1.18}, set
$$E = \bigcup_{q \in \mathbb{Q}\cap[0,1]} \{q\} = [0,1]\cap\mathbb{Q}$$
Then each $\{q\}$ for $q \in \mathbb{Q}\cap[0,1]$ is a Jordan measurable set, yet $E$ is not Jordan measurable. Similarly we see that
$$E = \bigcap_{q \in \mathbb{Q}\cap[0,1]} [0,1] \cap \{q\} =[0,1]\cap\mathbb{Q}$$
is not Jordan measurable as well.

DeeThreeCay · Answer

For union, let's consider
$$\bigcup_{q\in \mathbb{Q}\cap[0,1]} \{q\} = [0,1] \cap \mathbb{Q}$$
For intersection, let's consider
$$\bigcap_{q\in \mathbb{Q}\cap[0,1]} [0,1]\backslash \{q\} = [0,1] \cap \mathbb{I}$$

Exercise 1.2.1 (Countable unions/intersections of Jordan measurable sets need not be Jordan measurable)

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Exercise 1.2.1 (Countable unions/intersections of Jordan measurable sets need not be Jordan measurable)

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