请问这个多重积分的极限怎么求? 第1页

Let's take into account the generalized condition.

Let , is a continuous function, then, for any fixed , there holds

Proof:

Assuming a set of continuous random variables which are mutually independent follow an uniform distribution on the interval Owing to is continued for , therefore are also reciprocally independent random variables sequence for any fixed positive integer

Thus

Next, we will get the following formula via the Khinchin's law of large numbers

Hence

After that, according to the Lebesgue's Dominated Convergence Theorem, we have

Whereupon, we finally obtain

As for this question, let , we instantly get