Nonadditivity of Outer Measure for Disjoint Subsets

• categories: Measure Theory

Let $A$ and $B$ be two disjoint sets. The Outer Measure $m^{\ast }(\cdot )$ is not generally additive for disjoint sets, meaning:

$$m^{\ast }(A\cup B)\ne m^{\ast }(A)+m^{\ast }(B)$$

This occurs because the outer measure is based on coverings by arbitrary sets, not necessarily disjoint coverings. For two disjoint sets $A$ and $B$, the outer measure of their union may be strictly less than the sum of the outer measures of the individual sets.