Examples of Pointwise Convergence of Integrable Functions to Non-Integrable

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SUMMARY

The discussion centers around the identification of a sequence of integrable functions {fn} defined on the interval [0,1] that converges pointwise to a non-integrable function f. The proposed sequence is fn = 1/x for x > 1/n and fn = n for x < 1/n. A participant, mathman, challenges the pointwise convergence at x = 0, suggesting that redefining fn(0) to 0 resolves the issue. This highlights the nuances in defining convergence in the context of integrability.

PREREQUISITES
  • Understanding of pointwise convergence in real analysis
  • Familiarity with integrable functions and their properties
  • Knowledge of the Riemann integral
  • Basic concepts of sequences and limits in calculus
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Mathematicians, students of real analysis, and anyone studying the properties of integrable functions and convergence in mathematical analysis.

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hey guys,
Any one can think of any examples ?
Of a sequence of integrable functions{fn} on [0,1] that converges pointwise to a non-integrable function f:[0,1] --> R
??
 
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fn=1/x for x>1/n, fn=n for x<1/n.
 


mathman, I don't believe that function converges pointwise (specifically at 0). Setting fn(0) = 0 will fix that, though.
 

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