Interchange Summation and Integral

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Let’s explore the conditions under which we can rigorously interchange the summation and integral in the expression:

\ \int \sum f(x) \ dx = \sum \int f(x) \ dx \

1. Non-negativity of (f(x)):

If (f(x) \geq 0) for all (x) and for all (n),

then you can safely pull out the summation from the integral without any further conditions needed¹. This is a special case of the Fubini/Tonelli theorems.

2. Absolute Convergence: For general (f_n(x)), if either of the following conditions holds:
(\int \sum |f_n| dx < \infty)
(\sum \int |f_n| dx < \infty)

then you can interchange the summation and integral. These conditions are equivalent by Tonelli’s theorem. You can also prove this using the dominated convergence theorem¹.

In practice, these conditions cover most cases where you encounter such expressions.

The Dominated Convergence Theorem (DCT) is a powerful result in measure theory. It provides a sufficient condition under which limits and integrals of a sequence of functions can be interchanged. Here’s the statement:

Suppose we have a sequence of measurable functions (f_n: \mathbb{R} \to \mathbb{R}) that converge pointwise almost everywhere to a function f. If there exists an integrable function (g: \mathbb{R} \to [0,1]) such that (|f_n(x)| \leq g(x)) for all (n) and for all (x), then:

f \text{ is integrable, and } \int_{\mathbb{R}} f = \lim_{n \to \infty} \int_{\mathbb{R}} f_n

In other words, the integral of the limit function f is the limit of the integrals of the individual functions f_n The DCT is widely used in mathematical analysis, partial differential equations, and probability theory. It’s one of the key advantages of Lebesgue integration over Riemann integration¹². 😊📏

Source: Conversation with Copilot, 6/27/2024
(1) Dominated convergence theorem – Wikipedia. https://en.wikipedia.org/wiki/Dominated_convergence_theorem.
(2) Dominated Convergence Theorem – Math3ma. https://www.math3ma.com/blog/dominated-convergence-theorem.
(3) 지배 수렴 정리 – 위키백과, 우리 모두의 백과사전. https://ko.wikipedia.org/wiki/%EC%A7%80%EB%B0%B0_%EC%88%98%EB%A0%B4_%EC%A0%95%EB%A6%AC.
(4) Chapter 4. The dominated convergence theorem and applications. https://www.maths.tcd.ie/~richardt/MA2224/MA2224-ch4.pdf.

Source: Conversation with Copilot, 6/26/2024
(1) real analysis – When can a sum and integral be interchanged …. https://math.stackexchange.com/questions/83721/when-can-a-sum-and-integral-be-interchanged.
(2) calculus – relation between integral and summation – Mathematics Stack …. https://math.stackexchange.com/questions/184979/relation-between-integral-and-summation.
(3) Integration and Summation – MIT. https://www.mit.edu/~ehjin/files/Integration_And_Summation.pdf.
(4) Integration Rules – Math is Fun. https://www.mathsisfun.com/calculus/integration-rules.html.

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