why commutative integral with limit is important in real analysis?

Question

why commutative integral with limit is important in real analysis ?

Why $\lim_{n\to\infty }\int f_n=\int \lim_{n\to\infty } f_n $ is important ?

Answer 1

Because often we know how to compute $\int f_n$ for each individual $n$, but we don't know how to compute $\int\lim_{n\to\infty}f_n$ directly.

Answer 2

Certainly, what José Carlos Santos says is true. Often, we know how to compute $\int f_n$ for each $n$, but not how to compute $\int\lim_{n\rightarrow\infty}f_n$ directly. Exchanging limit and integration processes also has a lot of theoretical applications, which in turn have a practical implications.

Many more advanced theorems in measure theory are proven using such an interchange. Simple examples include differentiation under the integral or just continuity of certain functions defined by integrals (the prime example being the Gamma function). This also turns up in proving Fubini's Theorem and the Change of Variables Theorem, both of which are extremely useful in evaluating all kinds of integrals, which is something you often need to do in practice.

The interchange of limit and integral is also important in probability theory, where this often allows one to calculate the expectation of a limit of random variables. It is also integral (pun intended) in stating and proving the Radon-Nikodym Theorem, which is fundamental in understanding probability density functions. This also turns up in defining Conditional Expectations generally. These are both important tools in probability theory.

The list could go on and on. In conclusion, interchanging limit and integral is one of the most important techniques in developing measure theory, integration theory and probability theory, all of which are subjects both interesting in their own right, and with tons of applications.