Beispiel 3: Gleichverteilung/Maßtheorie

The set W := {t : ∃x t ∈ B_{x}} is a Σ_{1}^{1} set, hence measurable. Assume that μ(W) > 0. So there exists a Borel (closed) set A with μ(A) > 0 such that A ⊆ W. Note that the statement

(∗) ∀t : [t ∈ A ⇒ ∃x t ∈ B_{x}]

is a Π_{2}^{1}-statement, hence absolute between any two transitive universe with the same ordinals [Shoenfield].

Now let r be a random real over the universe V, r ∈ A, and work in V [r]. Random forcing is ℕ^{ℕ}-bounding, i.e., we have

∀x ∈ ℕ^{ℕ} ∃x′ ∈ ℕ^{ℕ} ∩ V : x ≤ x′

This implies ∀x ∈ ℕ^{ℕ} ∃x′ ∈ ℕ^{ℕ} ∩ V : B_{x} ⊇ B_{x′}, and so

⋃_{x∈ ℕ^{ℕ}} B_{x} = ⋃_{x′∈ ℕ^{ℕ}∩V} B_{x′}

Moreover,

∀x′ ∈ ℕ^{ℕ} ∩ V : r ∉ B_{x′}

(since r is a random real and each B_{x′} is of measure 0), so together we get in V [r]:

∀x ∈ ℕ^{ℕ} : r ∉ B_{x}.

Since (∗) holds also in V [r], we get V [r] ⊧ r ∉ A, a contradiction.