Htykuut

Let $[n] = {1 , dots, n}$ and let $mathcal{A} in mathcal{P}([n])$ be an increasing set, i.e. if $x in mathcal{A}$ and $x subset y subset [n]$ then $y in mathcal{A}$.

For $p in [0,1]$, define the $p$-biased measure of a subset $A subset [n]$ to be:

$$mu_p(A) = p^{|A|}(1-p)^{n - |A|}$$

For a family of sets $mathcal{A} subset mathcal{P}([n])$, define:

$$mu_p(mathcal{A}) = sum_{A in mathcal{A}} mu_p(A)$$

Theorem 7 says that for $0 < p <1$ and $mathcal{A}, mathcal{B} subset mathcal{P}([n])$ both increasing families of sets, then:

$$mu_p(mathcal{A})mu_p(mathcal{B}) leq mu_p(mathcal{A} cap mathcal{B})$$

With some work, I've managed to show that:

$$mu_p(mathcal{A})mu_p(mathcal{B}) = sum_{A in mathcal{A} backslash mathcal{B} \ B inmathcal{B} backslash mathcal{A}}{mu_p(mathcal{A})mu_p(mathcal{B})} +
sum_{C in mathcal{A} cap mathcal{B}}{mu_p(C) sum_{A in mathcal{A} backslash mathcal{B} \ B inmathcal{B} backslash mathcal{A}} (mu_p(A) + mu_p(B)}) + (mu_p(mathcal{A} cap mathcal{B})^2$$

However, at this point I am stuck and I'm not sure how to proceed, or how to use the fact that $mathcal{A}, mathcal{B}$ are increasing sets. How can I use this piece of information?