Conditional Measure and the Violation of Van Lambalgen’s Theorem for Martin-Löf Randomness

Abstract

Van Lambalgen’s theorem states that a pair (α, β) of bit sequences is Martin-Löf random if and only if α is Martin-Löf random and β is Martin-Löf random relative to α. In [Information and Computation 209.2 (2011): 183-197, Theorem 3.3], Hayato Takahashi generalized van Lambalgen’s theorem for computable measures P on a product of two Cantor spaces; he showed that the equivalence holds for each β for which the conditional probability P(⋅|β) is computable. He asked whether this computability condition is necessary. We give a positive answer by providing a computable measure for which van Lambalgen’s theorem fails. We also present a simple construction of a computable measure for which conditional measure is not computable. Such measures were first constructed by Ackerman et al. ([1]).

Appendix Two Definitions of Conditional Measure Coincide

In probability theory, conditional measures are defined implicitly using the Radon-Nikodym theorem. Any measure that satisfies the conditions of this theorem can be used as a conditional measure. The following lemma states that such measures are almost everywhere equal to the conditional measure P _C defined above.

Lemma 2 (Folklore)

Let 2 ^∗ be the set of strings. For every measure P on $2^{\mathbb {N}} \times 2^{\mathbb {N}}$ and for every function $f: 2^{*} \times 2^{\mathbb {N}}$ such that for all x and y

$$P([x],[y]) = {\int}_{[y]} f(x,\beta) P_{M}(\text{d}\beta), $$

we have that f(⋅,β)=P _C ([⋅]|β) for all β in a set of P _M -measure one.

In the proof we use the Lebesgue differentiation theorem for Cantor space. The proof of this version follows from the proof for Real numbers.

Theorem 6 (Lebesgue differentiation theorem for Cantor space)

Let Q be a measure on $2^{\mathbb {N}}$ . For every Q-integrable function $g:2^{\mathbb {N}} \rightarrow \mathbb {R}$ we have that

$$\lim_{n} \frac{{\int}_{[\beta_{1},\dots,\beta_{n}]} g(\gamma) Q(\text{d}\gamma)}{Q([\beta_{1},\dots,\beta_{n}])} = g(\beta) $$

for Q-almost all β.

Proof Proof of Lemma 2

For a fixed x, apply the Lebesgue differentiation theorem with g(⋅) = f(x,⋅) and Q = P _M . By assumption on f, the nominator simplifies to P([x],[β ₁…β _n ]). It follows that f(x, β) differs from P _C ([x]|β) in at most a set of β with P _M -measure zero. Because there are countably many strings x, it follows that f(⋅,β) and P _C ([⋅]|β) differ in at most a set of P _M -measure zero. □