Lebesgue Integrable Function question

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In summary, the problem is to show that for a Lebesgue integrable function f in the interval [a,b], the limit of the integral from a to b of (f(x)*|cosnx|) as n approaches infinity is equal to 2/pi times the integral from a to b of f(x). The attempt at a solution involves using the fact that every measurable function can be approximated by simple functions, and the special case of f being the characteristic function of an interval is suggested as a possible approach.
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TaylorWatts
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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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  • #2
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]?
 

What is a Lebesgue Integrable Function?

A Lebesgue Integrable Function is a type of mathematical function that can be integrated using the Lebesgue integral. This type of integral is used in measure theory to extend the concept of integration to a wider range of functions than the traditional Riemann integral.

What is the difference between a Lebesgue Integrable Function and a Riemann Integrable Function?

The main difference between a Lebesgue Integrable Function and a Riemann Integrable Function is the way they are integrated. A Riemann Integrable Function is integrated by dividing the interval into smaller subintervals and taking the limit as the subintervals approach zero. A Lebesgue Integrable Function, on the other hand, is integrated using the concept of measure, which considers the size and location of the set of points where the function is non-zero.

Why is the concept of Lebesgue Integrable Functions important?

The Lebesgue Integral extends the concept of integration to a wider class of functions, allowing for the integration of more complex and irregularly behaving functions. This is especially important in fields such as probability, where the use of Lebesgue Integrals is necessary to properly model and analyze random phenomena.

What are some properties of Lebesgue Integrable Functions?

Some properties of Lebesgue Integrable Functions include linearity, monotonicity, and absolute integrability. They also follow the Fatou's Lemma, which states that the integral of a sequence of functions is less than or equal to the limit of the integrals of the individual functions.

How can I determine if a function is Lebesgue Integrable?

In general, determining if a function is Lebesgue Integrable can be a complex task. However, there are some conditions that can be used to determine if a function is Lebesgue Integrable, such as being bounded, measurable, and having finite measure. Additionally, certain theorems, such as the Lebesgue Dominated Convergence Theorem, can be used to prove the integrability of a function.

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