Interchange of integrals and limits

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Discussion Overview

The discussion revolves around the interchange of integrals and limits, specifically exploring examples of convergent sequences of real-valued functions where the integral of the limit differs from the limit of the integrals. The scope includes theoretical aspects of analysis and convergence properties of functions.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant requests an example of a convergent sequence of functions where the integral of the limit is not equal to the limit of the integrals.
  • Another participant suggests that a sequence with non-uniform convergence could serve as an example.
  • A specific limit involving an integral is presented, indicating that the integral of the limit is zero while the limit of the integrals is one.
  • A question is raised about whether the same result would hold if the Lebesgue integral were applied to the sequence.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the example provided, and there is an ongoing exploration of the implications of different types of integrals.

Contextual Notes

The discussion does not clarify the assumptions regarding the type of convergence or the specific properties of the functions involved, which may affect the results presented.

Who May Find This Useful

Readers interested in real analysis, particularly in the properties of integrals and limits, as well as those studying convergence of sequences of functions.

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Can anybody please show me an example of a convergent sequence of real valued functions of a real variable where the integral of the limit of the sequence is different than the limit of the integrals of functions in the sequence ?

Thank you
 
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Try a sequence where convergence is not uniform
 
[tex]\lim _{n\rightarrow \infty } \left( \int _{0}^{1}\!{x}^{n} \left( 1-x<br /> \right) \left( n+1 \right) \left( n+2 \right) {dx} \right)[/tex]
 
So the integral of the limit is zero and the limit of the integrals is 1.
Thank you g_edgar
 
If we were taking the Lebesgue integral on this sequence, we would also have the same result, right?
 

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