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# Limit of a multiple integral

Prove that

Solution

Let be independent and uniform random variables. By the law of large numbers we have

in probability as . Therefore ,

in probability as .

The ratio random variables are bounded below by zero and above by one. This guarantees convergence of the expectations, as well.

So,

which is the required result.

Remark: In general it holds that

because the distribution of is the Lebesgue measure on hence for every measurable function ,