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Averaging a Kähler form over translations with normalized Haar measure gives an invariant Kähler form in the same de Rham cohomology class. Positivity is preserved by averaging positive values on $(v,Jv)$, and the class is unchanged because translations are homotopic to the identity. Thus any Hodge metric on a complex torus can be replaced by an invariant one.

For finite-dimensional rational representations of complex $G{L}_m$ or $S{L}_m$, average a Hermitian inner product over the compact group $U(m)$ or $SU(m)$ using normalized Haar measure. Orthogonal complements become invariant under the compact group and its complexified Lie algebra. The latter generates the complex group, giving invariant complements and hence complete reducibility. This conclusion does not hold for arbitrary affine algebraic groups, such as the additive group.

For a local field $F$, an additive character $\psi$, and a Haar measure $dx$, the Fourier transform isIt maps the Schwartz-Bruhat space to itself.

For a finite-dimensional continuous complex representation $R$ of a compact group, normalize Haar measure to mass one and average any positive-definite Hermitian inner product:The resulting inner product is positive-definite and invariant. If $W$ is an invariant subspace, then $W^{\perp }$ is invariant as well, since $⟨R(g)v,w{⟩}_K=⟨v,R(g^{-1})w{⟩}_K$. Thus $V=W\oplus W^{\perp }$. Restriction to a compact real form, followed by integration of a Lie-algebra representation, proves the Weyl complete reducibility theorem.

For an operator on a tensor-product Hilbert space, averaging unitary conjugations on subsystem $B$ with normalized Haar measure gives ${\mathcal{E}}_A(O)=({\mathrm{Tr}}_{B}O/d_B)\otimes I_B$. It fixes precisely the operators acting trivially on $B$ and is contractive in operator norm. Writing $O-UO{U}^{†}=[O,U]U^{†}$ converts localization error into a commutator bound.

For a continuous real representation $\rho :G\to GL(V)$ of a compact group, average any positive-definite real inner product against normalized Haar measure:The result is positive definite and $G$-invariant. An orthonormal basis for it conjugates the representation into an orthogonal group.