The need for measure theory

The problem with measure arises when one needs to decompose a body into (possibly uncountable) number of components and reassemble it after some action on those components.  Even when you restrict attention to just finite partitions, one still runs into trouble here. The most striking example is the Banach-Tarski paradox, which shows that a unit ball $B$ in three dimension can be disassembled into a finite number of pieces and reassembled to form two disjoint copies of the ball $B$.

Such pathological sets almost never come up in practical applications of mathematics.  Because of this, the standard solution to the problem of measure has been to abandon the goal of measuring every subset $E$ of $\mathbb{R}^d$ and instead to settle for only measuring a certain subclass of
non-pathological subsets of $\mathbb{R}^d$, referred to as the measurable sets.

The most fundamental concepts of measure is the properties of

1. finite or countable additivity
2. translation invariance
3. rotation invariance

The concept of Jordan measure (closely related to that of Riemann and Darboux integral) is sufficient for undergraduate level analysis.

However, the type of sets that arise in analysis, and in particular those sets that arise as limit of other sets, requires an extended concept of measurability (Lebesgue measurability)

Lebesgue theory is viewed as a completion of the Jordan-Darboux-Riemann theory.  It keeps almost all of the desirable properties of Jordan measure, but with the crucial additional property that many features of the Lebesgue theory are preserved under limits.