How to understand the operation "choose a random subset" in combinatorics?Define a set B ⊂ Z + randomly by requiring the events n ∈ B (for n ∈ Z + ) to be jointly independent with probability P ( n ∈ B ) = min ( C log ⁡ n n , 1 ) , 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 ( Ω , F , P ) here?2. Relating to the first question, how could I require that events { n ∈ B } n ∈ Z + to be jointly independent?3. Why could I require that events { n ∈ B } n ∈ Z + have the assigned probability?

Question

How to understand the operation &quot;choose a random subset&quot; in combinatorics?Define a set   B  ⊂            Z        +   randomly by requiring the events   n  ∈  B (for   n  ∈            Z        +  ) to be jointly independent with probability       P    (  n  ∈  B  )  =  min            (        C                    log        ⁡        n            n        ,  1            )      , where C  is a large constant to be chosen later.Since I&#039;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   (  Ω  ,      F    ,      P    ) here?2. Relating to the first question, how could I require that events   {  n  ∈  B      }          n      ∈                        Z                +             to be jointly independent?3. Why could I require that events   {  n  ∈  B      }          n      ∈                        Z                +             have the assigned probability?

thatuglygirlyu · Accepted Answer

Step 11. There may be several probability spaces. Example: when we speak about one coin toss, we may use   Ω  =  {  0  ,  1  },       F    =      2          Ω      ,   P  (  0  )  =  p,   P  (  1  )  =  1  −  p with   ξ  (  ω  )  =  ω or   Ω  =  [  0  ,  1  ],       F    =      B    [  0  ,  1  ], P-standard Lebesgue measure, with   ξ  (  ω  )  =      I          x      ∈      [      0      ,      p      ]      . Both of them are correct.In your problem we may put       Ω    1    =  {  (      i    1    ,      i    2    ,  …  )      |        i    j    ∈  {  0  ;  1  }  } and       F    =  σ-algebra containing the cylindrical sets. For example,   w  =  (  1  ,  0  ,  1  ,  1  ,  0  ,  …  ) means that   1  ∈  B  ,  2  ∉  B  ,  3  ∈  B  ,  4  ∈  B  ,  5  ∉  B  ,  … We also may put       Ω    2    =  [  0  ,  1      ]                  N             with corresponding   σ-algebra.Step 22-3. Put   a  (  n  )  =  min            (        C                    log        ⁡        n            n        ,  1            )       and define   P  (  w  ∈      Ω    1    :      w                  i        1              =  1  ,      w                  i        2              =  0  ,      w                  i        3              =  1  )  =      a                  i        1              (  1  −      a                  i        2              )      a                  i        3             allows us to define a measure on cylindrical   σ-algebra. The existence of such a measure follows from Kolmogorov existence theorem.Remark: the explicit form of   Ω is not important in such problems as it doesn&#039;t give any useful information.