Invariant integration

on a group

Integration of functions on a topological group that has a certain invariant property with respect to the group operations. Thus, let $G$ be a locally compact topological group, let $C _ {0} ( G)$ be the vector space of all continuous complex-valued functions with compact support on $G$ and let $I$ be an integral on $C _ {0} ( G)$, that is, a positive linear functional on $C _ {0} ( G)$( $I f \geq 0$ for $f \geq 0$). The integral $I$ is called left-invariant (or right-invariant) if $I ( gf ) = If$( or $I ( fg ) = If$) for all $g \in G$, $f \in C _ {0} ( G)$; here

$$( gf ) ( x ) = f ( g ^ {-} 1 x ) ,\ \ ( fg ) ( x) = f ( x g ) .$$

The integral $I$ is called (two-sided) invariant if it is both left- and right-invariant. The mapping $I \rightarrow \widehat{I}$, where $\widehat{I} f = I \widehat{f}$, $\widehat{f} ( x) = f ( x ^ {-} 1 )$, defines a one-to-one correspondence between the classes of left- and right-invariant integrals on $C _ {0} ( G)$. If $I = \widehat{I}$, then $I$ is called inversion invariant.

There exists on every locally compact group $G$ a non-zero left-invariant integral; it is unique up to a numerical factor (the Haar–von Neumann–Weil theorem). This integral is called the left Haar integral. The following equation holds:

$$I ( fg ) = \Delta ( g ) I f ,$$

where $g \in G$, $f \in C _ {0} ( G )$ and $\Delta$ is a continuous homomorphism from the group $G$ into the multiplicative group of positive real numbers (a positive character). Furthermore, $\widehat{I} f = I ( f / \Delta )$. The character $\Delta$ is called the modulus of $G$. If $\Delta ( g) \equiv 1$, then $G$ is called unimodular. In this case $I$ is a two-sided invariant integral.

In particular, every compact group (where $I1 < \infty$, $\widehat{I} = I$) and every discrete group (where $If = \sum _ {g} f ( g)$, $f \in C _ {0} ( G)$) is unimodular.

According to the Riesz theorem, every integral on $C _ {0} ( G)$ is a Lebesgue integral with respect to some Borel measure $\mu$ which is uniquely defined in the class of Borel measures that are finite on each compact subset $K \subset G$. The left- (or right-) invariant measure $\mu$ corresponding to the left (right) Haar integral on $C _ {0} ( G)$ is called the left (right) Haar measure on $G$.

Let $H$ be a closed subgroup of $G$ and let $\Delta _ {0}$ be the modulus of $H$. If $\Delta _ {0}$ can be extended to a continuous positive character of $G$( cf. Character of a group), then there exists on the left homogeneous space $X = G / H$ a relatively invariant integral $J$, that is, a positive functional on the space $C _ {0} ( X)$ of continuous functions with compact support on $X$ that satisfies the identity

$$J ( gf ) = \delta ( g ) J f$$

for all $g \in G$, $f \in C _ {0} ( X)$; here

$$( gf ) ( x) = f ( g ^ {-} 1 x ) ,\ \ \delta ( g) = \frac{\Delta _ {0} ( g) }{\Delta ( g) } ,$$

and $\Delta$ is the modulus of $G$. This integral is defined by the rule $J f = I ( \delta \widetilde{f} )$, where $I$ is the left Haar integral on $G$ and $\widetilde{f}$ is a function on $G$ such that

$$f ( gH ) = I _ {0} (( g \widetilde{f} ) _ {H} ) .$$

( $I _ {0}$ is the left Haar integral on $H$ and $\phi _ {H}$ is the restriction of $\phi$ to $H$.) This is well-defined since $\widetilde{f} \rightarrow f$ is a mapping from $C _ {0} ( G)$ onto $C _ {0} ( X)$ and $Jf = 0$ when $f = 0$. The notion of an invariant mean (cf. Invariant average) is closely related to that of invariant integration.

References

 [1] N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) [2] A. Weil, "l'Intégration dans les groupes topologiques et ses applications" , Hermann (1940) [3] L.H. Loomis, "An introduction to abstract harmonic analysis" , v. Nostrand (1953) [4] E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1 , Springer (1979)