Editing Gauss–Kuzmin–Wirsing operator (section)

=== Measure-theoretic preliminaries ===
A covering family <math>\mathcal C</math> is a set of measurable sets, such that any open set is a ''disjoint'' union of sets in it. Compare this with [[Base (topology)|base in topology]], which is less restrictive as it allows non-disjoint unions.

'''Knopp's lemma.''' Let <math>B \subset [0, 1)</math> be measurable, let <math>\mathcal C</math> be a covering family and suppose that <math>\exists \gamma > 0, \forall A \in \mathcal C, \mu(A \cap B) \geq \gamma \mu(A)</math>. Then <math>\mu(B) = 1</math>.

'''Proof.''' Since any open set is a disjoint union of sets in <math>\mathcal C</math>, we have <math>\mu(A \cap B) \geq \gamma \mu(A)</math> for any open set <math>A</math>, not just any set in <math>\mathcal C</math>.

Take the complement <math>B^c</math>. Since the Lebesgue measure is [[Regular measure|outer regular]], we can take an open set <math>B'</math> that is close to <math>B^c</math>, meaning the symmetric difference has arbitrarily small measure <math>\mu(B' \Delta B^c) < \epsilon</math>.

At the limit, <math>\mu(B' \cap B) \geq \gamma \mu(B')</math> becomes have <math>0 \geq \gamma \mu(B^c)</math>.