A correct statement could be: Let be a family of independent random variables uniformly distributed on . What can be said about the (random variable) , if this is well-defined?

As a matter of fact, I would bet that is almost-surely not measurable, hence the integral wouldn't make sense.

You can try another way, for instance as an approximation: for , define (or with a,b,c,d) to be a step function where the steps have width and independent uniformly distributed heights in . This won't converge to a function when . However, almost-surely by the law of large numbers, thus for large this function nearly satisfies what you said.

By the way, there is a name for such a function : it is called a white noise, you can look that up.