Family of all elementary subsets of $\mathbb{R}^n$ is a ring, but not a $\sigma$-ring

Question

I was going through the introduction to Lebesgue theory in Baby Rudin, where the property was given that:

$\mathcal{E}$ is a ring, but not a $\sigma$-ring.

$\mathcal{E}$ here represents the family of all elementary subsets of $\mathbb{R}^n$, $n$-dimensional Euclidean space.

An elementary set is the union of a finite number of intervals, where an interval is defined to be a set of points $(x_1,\dots,x_n)$ in $\mathbb{R}^n$ such that

$$ a_i \leq x_i \leq b_i ~~~~~(i=1,\dots,n). $$

I am having trouble understanding why $\mathcal{E}$ does not satisfy the properties of a $\sigma$-ring.

I apologize if the question is very trivial, the notion of interval $I$ and $m(I)$ of $I$ is very new and abstract to me.

Answer 1

The reason is simple:

Finite union of elementary sets is always elementary set (union of a finite number of intervals), but countable union of elementary sets may not be elementary set anymore. It is easy to find examples.