Invariant Measure - Examples

Examples

• Consider the real line R with its usual Borel σ-algebra; fix a ∈ R and consider the translation map T_a : R → R given by:

Then one-dimensional Lebesgue measure λ is an invariant measure for T_a.

• More generally, on n-dimensional Euclidean space Rn with its usual Borel σ-algebra, n-dimensional Lebesgue measure λn is an invariant measure for any isometry of Euclidean space, i.e. a map T : Rn → Rn that can be written as

for some n × n orthogonal matrix A ∈ O(n) and a vector b ∈ Rn.

• The invariant measure in the first example is unique up to trivial renormalization with a constant factor. This does not have to be necessarily the case: Consider a set consisting of just two points and the identity map which leaves each point fixed. Then any probability measure is invariant. Note that S trivially has a decomposition into T-invariant components {A} and {B}.

• The measure of circular angles in degrees or radians is invariant under rotation. Similarly, the measure of hyperbolic angle is invariant under squeeze mapping.

• Area measure in the Euclidean plane is invariant under 2 × 2 real matrices with determinant 1, also known as the special linear group SL(2,R).

• Every locally compact group has a Haar measure that is invariant under the group action.