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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 $$\lim_{n\rightarrow\infty}\int_a^bf_n(x)dx=\int_a^b\lim_{n\rightarrow\infty}f_n(x)$$

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.

Ibrahim Islam
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Some problems I noticed when studying Riemann's integral that disappear with the Lebesgue integral:

The Riemann integral can only integrate functions $f \colon \mathbb{R}^n \to \mathbb{R}$ 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.

Mason
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  • ;but my question is still not answered,I want to know why do we bother about interchange of limit and integral sign?what's important in that? I heard that this will help convergence of Fourier series,but I don't know how,some says it will be helpful while working with complete space $L^p$ but again I have no idea how....can you please elaborate if you know any such points? – Ibrahim Islam Nov 03 '21 at 15:33
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    @lbs Some examples of when it is useful switch integral and limit: 1. Interchanging partial derivative and integral. This comes up in Fourier analysis for proving $\partial_j F(f) = F(-ix_jf)$ for $f \in S(\mathbb{R}^n)$. 2. Ability to get general theorems by proving special cases on small class of functions. Many proofs in measure theory prove the theorem first for a small class of functions like $C_c^{\infty}(\mathbb{R}^n)$, or simple functions and then use limits to get the theorem for all measurable $f \geq 0$ and all $f \in L^1$. – Mason Nov 03 '21 at 22:25