Solving Probability Integrals with Monotone Convergence Theorem

[shoeburg]

I'm having trouble working out a few details from my probability book. It says if P(An) goes to zero, then the integral of X over An goes to zero as well. My book says its because of the monotone convergence theorem, but this confuses me because I thought that has to do with Xn converging to X. Here is my attempt anyway:

We can write the integration as lim n to infinity of E[X(I(An))] = E(lim n to infinity of X P(An)) = E(X(0)) = E(0) = 0.

 

[mathman]

It would be helpful if you defined the symbols. An, X, Xn, etc.?

 

[shoeburg]

Let An be a sequence of sets indexed by n. As n goes to infinity, P(An) = 0. Let X be a random variable. Prove that the limit as n goes to infinity of the integral of X over An, with respect to probabiliity measure P, equals zero. I thought that the monotone convergence theorem applies to when a sequence of random variables Xn converges to X, and you can interchange limit taking and integral taking (expectation). How does it apply here?

Thanks.

 

[verty]

This may be totally wrong, but if the probability of any $A_i$ is greater than 0, then surely the integral is greater than 0, being the probability of a superset? I suppose this must be wrong but it is the most obvious way to interpret the question.

 

[shoeburg]

I do agree that the integral of anything over a set of measure zero is zero. However, if X takes the value 0 over a set B with P(B) > 0, than the expectation (integral) of X over that set B is zero. The concept that an integral over a set with probability zero is zero is certainly intuitive, it is the details of the proof that I am wondering about, namely, when can you apply the limit and let n go to infinity? Surely, we cannot just suddenly apply the limit immediately, but must calculate the integral somehow, perhaps how I tried it posted above.

 

[mathman]

The "theorem" is false. Example:

Let X be a random variable with a Cauchy distribution. An = set of points where X > n. P(An) -> 0. However the integral of X over An is infinite for all n.

 

[shoeburg]

You are right. Supposing I said, assume E(X) is finite. Does that take care of this counterexample and issue?

 

[Office_Shredder]

When you say the integral of X over A_n, do you mean
$$\int_{A_n} x p(x) dx$$
or
$$\int_{A_n} p(x) dx$$

where p(x) is the distribution?

 

[shoeburg]

The first one, I believe. My book usually has it written as X dP, but what you have written in the first one is equivalent, correct? And then you just put the limit taking as n -> infinity before the integral.

An = set of points where X > n. P(An) -> 0.

How do you know P({X>n}) --> 0? Is this true for all random variables with a finite expectation?