How to understand the operation "choose a random subset" in combinatorics?Define a set B ⊂...
How to understand the operation "choose a random subset" in combinatorics?
Define a set randomly by requiring the events (for ) to be jointly independent with probability , where C is a large constant to be chosen later.
Since I've not seen such a method before, I have several questions:
1. If one talks about randomness, then there should be a probability space. In the proof they choose a set B randomly, what is the probability space here?
2. Relating to the first question, how could I require that events to be jointly independent?
3. Why could I require that events have the assigned probability?
Answer & Explanation
1. There may be several probability spaces. Example: when we speak about one coin toss, we may use , , , with or , , P-standard Lebesgue measure, with . Both of them are correct.
In your problem we may put and -algebra containing the cylindrical sets. For example, means that We also may put with corresponding -algebra.
2-3. Put and define allows us to define a measure on cylindrical -algebra. The existence of such a measure follows from Kolmogorov existence theorem.
Remark: the explicit form of is not important in such problems as it doesn't give any useful information.