In general, no. But the famous convergence theorems of Lebesgue give you some conditions under which the answer is yes. The one that you want here is Lebesgue's monotone convergence theorem.
Hello everyone I was wondering if you could help me with something. Let for where and chi is the characteristic function. Then f is measurable since it is the limit pointwise (almost) everywhere of step functions, but how do I show that ?
If I define , then of course
.
But since the limit of this sequence is infinity, does this tell me that the integral of f diverges? Can I say that the integral of a measurable function ?
The definition of the integral of a non-negative measurable function is that you take the truncation if (if ) and leave it the same otherwise, then
so in the context of my question I guess I am trying to show that
,
but that seems like a giant mess I can't make sense of. Can anyone give me a hand with this. Thanks a lot
In general, no. But the famous convergence theorems of Lebesgue give you some conditions under which the answer is yes. The one that you want here is Lebesgue's monotone convergence theorem.