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