Question

Prove that every set A of the outer measure 0

∀Z⊆R∣Z∩A∣+∣Z\A∣=∣Z∣ is equivalent to: ∣Z1​∣+∣Z2​∣=∣Z1​∪Z2​∣ for any sets Z1​⊆A and Z2​⊆R\A.

Answer 1

The statement is not generally true. While the equation holds for some sets A with outer measure 0, it does not hold for all such sets.

Step-by-step explanation:

The statement claims that for any set A with outer measure 0, and any two subsets Z1 and Z2, both contained within A and its complement respectively, the sum of their measures equals the measure of their union. While this may be true for some special cases, it does not hold as a general rule.

Here's why:

Outer measure 0 does not imply countable disjointness: A set with outer measure 0 can still be decomposed into a countable number of non-empty sets, violating the necessary condition for additivity of measures (finite disjoint unions).

The statement relies on specific subsets: The equation only applies to Z1 within A and Z2 within its complement. If either subset overlaps the other or the original set A, the additivity principle would not hold.

Therefore, while the statement may hold for specific cases of A with outer measure 0 and carefully chosen subsets, it lacks generality and cannot be proven true for all such sets.

For a complete proof, counterexamples showcasing sets with outer measure 0 that violate the stated equation would be required.