Proving a true fact about measure theory and integration

• jdinatale

jdinatale

So the above is the problem and my idea of how to approach it. This problem comes from the section on the Countable Additivity of Integration and the Continuity of Integration, but I was not sure how to incorporate those into the prove, if you even need them for the result.

I had no idea what to do after Case 1, leading me to believe that that approach is wrong.

Case 1 is just silly - all you did was observe that if you integrate over an empty set you get zero. Don't think that there's any way to generalize it to the unbounded case.

Countable additivity is one definite option for solving this problem. Your Ens are not disjoint, but can you think of a way to write down some disjoint sets whose unions can give you the various Ens?

So the above is the problem and my idea of how to approach it. This problem comes from the section on the Countable Additivity of Integration and the Continuity of Integration, but I was not sure how to incorporate those into the prove, if you even need them for the result.

I had no idea what to do after Case 1, leading me to believe that that approach is wrong.

The problem is telling me that I want to prove that the sequence $\left(\int_{E_n} f\right)_{n \geq 0}$ converges to 0 as $n \to \infty$.

Hint: Find a telescoping series whose limit is $\int_E f$, and look at the sequence of partial sums.