Orthogonally Invariant Probability Measure

• Orthogonally invariant probability measure is defined as a measure that remains unchanged under isometries, including rotations and reflections, on compact metric spaces.
• The construction utilizes covering numbers and an ultrafilter-based limit, ensuring uniqueness and compatibility with classical Lebesgue measure for Euclidean domains.
• These measures are essential in applications such as random matrix theory, ergodic theory, and statistical models requiring orthogonal invariance.

An orthogonally invariant probability measure is a probability measure defined on a metric space (often but not exclusively a Euclidean space, a matrix space, or a more general compact metric space) such that the measure is invariant under the action of the orthogonal group, i.e., under all isometries (distance-preserving maps) that include rotations and reflections. This concept generalizes the classical Lebesgue measure and underpins many key developments in geometry, probability, and mathematical physics, especially in contexts requiring statistical or physical invariance under rotations. The rigorous construction and analysis of such measures address existence, uniqueness, invariance properties (including under partial or local isometries), and connections with both classical and modern functional analysis, probability theory, and geometry.

1. Construction of Orthogonally Invariant Probability Measures

Given a nonempty compact metric space $K$, the construction process for an orthogonally invariant measure proceeds without assuming a global group structure but instead exploits partial isometries on open subsets. For a subset $A \subset K$, define the minimal covering number $N(A,\epsilon)$ as the smallest number of open balls of radius $\epsilon$ needed to cover $A$. Then set

$$\mu_K(A) = L\left( \frac{N(A, \epsilon)}{N(K, \epsilon)} \right),$$

where $L$ is an ultrafilter-based limit as $\epsilon \to 0$. The limit notion $L$ resolves potential non-uniqueness in the limit and guarantees regularity.

Next, define for $P \subset K$,

$$m'(P) = \sup \left\{ \mu_K(U) \mid U \subset P,\, U \in M \right\},$$

where $M$ is an algebra of “good” sets (typically closed or open subsets closed under unions and intersections). Finally, regularize via open covers: $$p(P) = \inf\left\{ \sum_{i \in I} m'(U_i) \mid P \subset \bigcup_{i \in I} U_i,\, U_i \text{ open} \right\}.$$ This yields a metric outer measure whose restriction to Borel sets is a probability measure. The construction parallels the classical Hausdorff measure but is adapted to arbitrary compact metric spaces and respects partial isometries (Larrieu, 2011).

2. Invariance Properties and Orthogonal Compatibility

A measure $p$ is isometrically compatible if, for any Borel set $A$ in an open set $U$ and any partial isometry $f: U \to V$ (an isometry between open subsets), $p(f(A)) = p(A)$. Invariance arises because the construction uses only covering numbers, which are preserved under isometries: $$\frac{N(f(A), \epsilon)}{N(K, \epsilon)} = \frac{N(A, \epsilon)}{N(K, \epsilon)},$$ hence $p(f(A)) = p(A)$.

On homogeneous spaces—those where for any $x, y \in K$ there exists a local isometry mapping a neighborhood of $x$ to a neighborhood of $y$—the measure is unique up to scaling and, when normalized, unique among all isometrically compatible probabilities.

A probability measure is called metrically compatible if it is invariant under all $1$-Lipschitz (and $1$-coercive) maps. The construction in (Larrieu, 2011) guarantees the existence of a measure which is at least isometrically compatible on any compact metric space.

In Euclidean settings, isometric and metrically compatible invariance ensure invariance under the entire orthogonal group $O(d)$; in more general spaces, the invariance is local or “partial.”

3. Uniqueness and the Generalization of the Lebesgue Measure

On compact homogeneous metric spaces, the constructed orthogonally invariant probability measure is unique. In the classic case where $K \subset \mathbb{R}^d$ is a compact set with non-empty interior, the construction recovers (after normalization) the Lebesgue measure: $p_K(A) = \frac{\lambda(A)}{\lambda(K)},$ where $\lambda$ is the Lebesgue (volume) measure on $\mathbb{R}^d$. Thus, the measure is a genuine metric generalization of Lebesgue measure.

For products, the probability on $K \times K'$ is $p_K \otimes p_{K'}$—the product measure of the orthogonally invariant probabilities on the factors. This is a direct analog to Fubini’s theorem for Lebesgue measure (Larrieu, 2011).

4. Existence: Krylov–Bogoliubov-Type Equivalence Theorem

The paper generalizes the classic existence theory for invariant probability measures, akin to the Krylov–Bogoliubov approach. For a compact monoid $M$ of measurable maps on $K$ (with a right-invariant pseudometric),

$D(f,g) = \int_K d(f(x), g(x))\, dp(x),$

there exists an invariant probability measure on $M$, and hence on $K$. This allows broad existence results in far-reaching generality—not requiring a group action, but only a suitably compact semigroup/monoid with metric compatibility (Larrieu, 2011).

For example, if $M$ is the set of Borel maps satisfying

$p(f^{-1}(A)) = p(A),\quad \forall A \subset K,$

the compactness (in the metric $D$) of $M$ ensures the existence of an invariant probability on $M$ itself, and, via averaging, on $K$.

5. Relation to Orthogonally Invariant Probability Measures and Applications

The construction yields a measure that is invariant under all possible local isometries, thereby generalizing “orthogonal invariance” as customarily understood in matrix analysis, random matrix theory, statistics, and geometry. In matrix or vector spaces, invariance under the group $O(d)$ exactly coincides with orthogonal invariance.

The constructed measure supports key properties required in applications:

• Recovering Lebesgue measure on Euclidean domains,
• Ensuring canonical uniqueness on homogeneous compact metrics spaces,
• Full product structure under Cartesian powers of base spaces,
• Extending the reach of the Krylov–Bogoliubov theorem to monoids and without assuming global group structure,
• Providing a measure-theoretic foundation for rotationally or orthogonally invariant statistical models in high-dimensional probability and geometric analysis.

This framework is foundational for problems in stochastic geometry, ergodic theory, optimal transport, random matrix theory, and statistics where invariance is essential, and for constructing reference measures on metric spaces where group symmetries may act only locally (Larrieu, 2011).

6. Key Formulas and Summary Table

The principal constructs and invariance properties are summarized in the table:

Concept	Definition / Formula	Note
$\mu_K(A)$ (candidate measure)	$$\displaystyle \mu_K(A) = \lim_{\epsilon \to 0} \frac{N(A,\epsilon)}{N(K,\epsilon)}$$	Limit via ultrafilter $L$
Regularization and outer measure	$$\displaystyle m'(P) = \sup \{\mu_K(U): U \subset P,\, U\in M\}$$<br>$$\displaystyle p(P) = \inf \left\{\sum m'(U_i): P\subset \bigcup U_i,\, U_i~\text{open}\right\}$$	$p$ is a metric outer measure
Invariance under partial isometries	$p(f(A)) = p(A)$ (for $f:U \to V$ partial isometry)	Isometric or metrically compatible
Lebesgue measure recovery	$p_K(A) = \lambda(A)/\lambda(K)$	When $K\subset\mathbb{R}^d$, translation-invariant metric
Krylov–Bogoliubov equivalence	Existence of invariant probability for compact monoids $M$ of maps with right-invariant pseudometric $D$	See discussion above

This approach provides both the existence and uniqueness of orthogonally invariant measures and supports applications where invariance under isometries or orthogonal transformations is a fundamental structural property.