Invariant

A property or quantity is invariant under a transformation if it remains unchanged when that transformation is applied.

Definition

Let $T: X \to X$ be a transformation on a set $X$. A subset $A \subseteq X$ is invariant under $T$ if:

$$ T(A) = A $$

More generally, a function $f: X \to Y$ is invariant under $T$ if:

$$ f(T(x)) = f(x) \quad \text{for all } x \in X $$

Examples

1. Translation invariance: A measure $\mu$ on $\mathbb{R}^n$ is translation invariant if $\mu(A + x) = \mu(A)$ for all measurable sets $A$ and all $x \in \mathbb{R}^n$. Lebesgue measure has this property.

2. Rotation invariance: A function $f: \mathbb{R}^2 \to \mathbb{R}$ is rotation invariant if $f(x, y)$ depends only on $\sqrt{x^2 + y^2}$.

3. Group invariants: In group theory, an invariant is a quantity that remains unchanged under the action of a group.

Related

• measure
• automorphism