What is the condition for a function to be integrable with improper integral?

[LHeiner]

Hello

i stumbled over a question and I'm not sure how to proof/ solve hat:

At how many points a function can not be defined to be nevertheless integrable (improper integral)?

Thx in advance!

 

[mathman]

If you are talking about Riemann integral, a countable number of points. However in many cases there could be a problem, since the set of rationals is countable.

For Lebesgue integral, a set of measure 0. This is a much better criterion.

 

[disregardthat]

Interestingly for lebesgue integrals a function can be left undefined on an uncountable subset of R and still be integrable.

An even weaker condition is weaker for lebesgue integrals though, the set need only be a subset of a measurable set of measure 0.

For riemann integrals I think it is sufficient that the countable set of points where the function is undefined is not dense in any interval (a,b) of R.