# I just started studying measure theory.The introduction starts with the failure of limit of integral

I just started studying measure theory.The introduction starts with the failure of limit of integral not equal to integral of limit in Riemann integration.I want to know why this problem
$\underset{n\to \mathrm{\infty }}{lim}{\int }_{a}^{b}{f}_{n}\left(x\right)dx={\int }_{a}^{b}\underset{n\to \mathrm{\infty }}{lim}{f}_{n}\left(x\right)$

is so important in analysis or what are benefits of limit of integral being equal to integral of limit.
Also, is this only drawback of Riemann integral?Is there any other problem with Riemann integration which leads to generalisation of Riemann to lebesgue integration?please clearly explain the motivation behind lebegue integral so that I could develop intrest in subject.
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Zackery Harvey
Some problems I noticed when studying Riemann's integral that disappear with the Lebesgue integral:
The Riemann integral can only integrate functions $f:{\mathbb{R}}^{n}\to \mathbb{R}$ with bounded support and bounded range, though the Riemann integral can be extended to integrate certain functions with unbounded support and unbounded range.
Not all compact sets are Riemann measurable. Countable unions of Riemann measurable sets are not necessarily measurable. A pointwise limit of Riemann integrable functions is not necessarily Riemann integrable.