Computing the probability of intersection

Let normal upper Omega 1 comma ellipsis comma normal upper Omega Subscript m Baseline Ω 1 , … , Ω m Ω ₁, … , Ω ₘ be probability spaces, let bold upper Omega equals normal upper Omega 1 times midline horizontal ellipsis times normal upper Omega Subscript m Ω = Ω 1 × ⋯ × Ω m Ω =Ω ₁ × ⋯ × Ω ₘ be their product and let upper A 1 comma ellipsis comma upper A Subscript n Baseline subset of bold upper Omega A 1 , … , A n ⊂ Ω A₁, … , Aₙ ⊂ Ω be events. Suppose that each event upper A Subscript i A i Aᵢ depends on r Subscript i r i rᵢ coordinates of a point x element of bold upper Omega x ∈ Ω x ∈ Ω , x equals left parenthesis xi 1 comma ellipsis comma xi Subscript m Baseline right parenthesis x =