The [Kelly criterion](https://en.wikipedia.org/wiki/Kelly_criterion) gives a simple formula to calculate the **fraction** of one's current wealth/bankroll.

On the page above it says

> Assuming that the expected returns are known, the Kelly criterion leads to higher wealth than any other strategy in the long run (i.e., the theoretical maximum return as the number of bets goes to infinity).

Down below in article also, there is a heuristic argument given, saying that it maximizes some expected value. The heuristic argument of course remains vague.

To me the argument remains unconvincing, since obviously the set of all strategies is uncountable (i.e. at each step choose a real number between 0 and current wealth) and a natural measure on that set, does not appear to be obvious.


Now the question:
1. Is there a simple reason or heuristic argument that explains why there should even be a **unique** single strategy out of the uncountably many that is optimal? A nice compactness argument would make me happy.
2. Assuming 1), is there a simple reason or heuristic argument that explains that Kelly's criterion takes such a simple form - namely a constant factor of the wealth rather than some other more complicated function of wealth.