Lebesgue Integrable Function question

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SUMMARY

The discussion centers on proving that for a Lebesgue integrable function f over the interval [a,b], the limit of the integral of f(x)*|cos(nx)| as n approaches infinity equals (2/π) times the integral of f(x) over the same interval. The participant has successfully demonstrated a related case involving f*cos(nx) resulting in an integral of zero. They seek guidance on selecting an appropriate simple function for the current problem and inquire about the validity of the proof when f is the characteristic function of a subset [r,s] within [a,b].

PREREQUISITES
  • Understanding of Lebesgue integrable functions
  • Familiarity with the properties of measurable functions
  • Knowledge of simple functions and their approximations
  • Basic concepts of Fourier analysis, specifically the behavior of cos(nx)
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  • Research the properties of Lebesgue integrable functions and their implications
  • Study the approximation of measurable functions by simple functions
  • Explore the characteristics of the Fourier series, particularly in relation to integrals
  • Investigate the use of characteristic functions in integration and limit proofs
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Mathematics students, particularly those studying real analysis and integration theory, as well as educators looking to enhance their understanding of Lebesgue integration and Fourier analysis.

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Homework Statement



Let f be a Lebesgue integrable function in the interval [a,b]

Show that:

lim integral from a to b (f(x)*|cosnx|) = 2/pi * integral from a to b (f(x))
n->infinity

Homework Equations



Every measurable function can be approximated arbitrarily close by simple functions.

The Attempt at a Solution



I've solved part i of the problem (which was to show the same setup except f*cosnx instead of f*|cosnx|) has an integral equal to zero.

I'm pretty stuck on this part - I'm sure I have to find a simple function but I'm not sure what a good simple function would be.
 
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Can you prove this is true for the special case when f is the characteristic function of an interval I=[r,s], where [r,s] is a subset of [a,b]?
 

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