# Translational invariance

For translationally invariant functions is . This applies, for example, to the Lebesgue measure.${\ displaystyle f \ colon \ mathbb {R} ^ {2} \ rightarrow \ mathbb {R}}$${\ displaystyle f (A) = f (A + t)}$
The smaller relation on the real numbers is translation invariant.

As translationally invariant in are mathematics figures referred whose value is derived from a translation does not change. More specifically, means a functional translation-invariant, when the value of the functional does not change when the function of a translation with displacement vector is subjected: . ${\ displaystyle F (f) \ to \ mathbb {R}}$${\ displaystyle f \ colon \ mathbb {R} ^ {n} \ to \ mathbb {R}}$${\ displaystyle a \ in \ mathbb {R} ^ {n}}$${\ displaystyle Tf (x) = f (xa)}$

For example, every constant function is translation invariant. A more interesting example is the Lebesgue integral . Its translation invariance clearly means that the value of an integral does not change when the domain of definition is shifted, just as the volume of a body does not change through pure displacement in space.

## General definition: translational invariance in groups

More generally, it is possible to define translation invariance in group operations. Let X a set with a transitive operation of a group G . Then induced

${\ displaystyle x \ to gx}$

for each element g of G an automorphism of X and thus an automorphism on each functorial structure F (X) on X . The G invariants in F (X) are called translation invariant.

For a group G and X = G one can through

${\ displaystyle h \ to gh}$ and ${\ displaystyle h \ to hg ^ {- 1}}$

Define two G-spaces, the associated translation invariance is called left or right invariance.

For example, the Lie algebra of a Lie group is the space of the left invariant vector fields . A Haar measure on a topological group is also translation-invariant. The Petersson scalar product on the upper half-plane is defined with the help of an SL (2, R) -invariant measure.

## Others

Translation invariant is also a stochastic function that is only changed by additive (or subtractive) components. The laws that are described with the function are not affected. Only the mean or scale values change.