Examples of Pointwise Convergence of Integrable Functions to Non-Integrable

Thread starter irresistible
Start date Feb 9, 2009
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In summary, pointwise convergence of integrable functions to non-integrable refers to a sequence of functions approaching a non-integrable function at every point in a given domain. This concept is important for understanding the behavior and properties of the non-integrable function. It is possible for a sequence of integrable functions to converge pointwise to a non-integrable function, such as the example of f_n(x) = x^n on the interval [0,1]. Pointwise convergence is different from uniform convergence in that it only requires convergence at every point in the domain, while uniform convergence requires uniform convergence on the entire domain.

Feb 9, 2009

irresistible

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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Feb 10, 2009

mathman

Science Advisor

fn=1/x for x>1/n, fn=n for x<1/n.

Feb 11, 2009

adriank

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

Related to Examples of Pointwise Convergence of Integrable Functions to Non-Integrable

What is pointwise convergence of integrable functions to non-integrable?

Pointwise convergence refers to the concept of a sequence of functions approaching a specific function at every point in a given domain. In the case of integrable functions, this means that the sequence of functions approaches a non-integrable function at every point in the domain.

Why is pointwise convergence of integrable functions to non-integrable important?

This concept is important because it allows us to study the behavior of a sequence of functions and understand how it approaches a non-integrable function. It also helps us to determine the properties of the non-integrable function in terms of its integrability.

Can a sequence of integrable functions converge pointwise to a non-integrable function?

Yes, it is possible for a sequence of integrable functions to converge pointwise to a non-integrable function. This means that at every point in the domain, the sequence of functions approaches the non-integrable function.

What are some examples of pointwise convergence of integrable functions to non-integrable?

One example is the sequence of functions defined by f_n(x) = x^n on the interval [0,1]. This sequence converges pointwise to the function f(x) = 0, which is not integrable on the given interval.

How is pointwise convergence of integrable functions to non-integrable different from uniform convergence?

Pointwise convergence is different from uniform convergence in that it only requires that the sequence of functions approaches the non-integrable function at every point in the domain, whereas uniform convergence requires that the sequence of functions approaches the non-integrable function uniformly on the entire domain.