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.
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.
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.
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.
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.