I am reading ''An Introduction to Measure Theory'' written by Terence Tao.
The use of a notion $m(E)$ is not comprehensible to me. The following is details
$\mathcal{E}$:A set of the elementary sets
$J$:A set of the Jordan measurable sets
$m:\mathcal{E}\to[0,\infty]$ :elementary measure
$m_J:J\to[0,\infty]$ :Jordan measure
By results written in this textbook, $\mathcal{E}\subseteq J$ and $m(E)=m_J(E)$ for all elementary sets $E$ are followed.
Hence, $m(E)$ is used to denote Jordan measure $m_J$.
I do not understand the justification of the use of $m(E)$ to denote elementary measure and Jordan measure.
I wish you answer to this question.
The answer to this question has nothing to do with measure theory per se. You seem to have some kind of misunderstanding regarding the concept of generalisation.
As you said, Jordan measure generalises elementary measure. This is similar to how real numbers generalise integers, so you do not use the latter set at all (you almost always use the integer subset of reals and not the integers themselves). In other words, if you were to be very pedantic, you would write $1_\mathbb{Z}$ and $1_\mathbb{R}$ when referring to 1 from the set of integers and the set of reals respectively (which are technically not the same), yet the latter generalises the former and you can therefore safely disregard the first notation and always use the "real" version of 1.
Later in this exercise, you will also see that Lebesgue measure generalises Jordan measure (like complex number generalise real numbers), so you will then switch to Lebesgue measure completely.
I hope this analogy clears the misunderstanding.