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G G/H in .NET Integration Code-39 in .NET G G/H




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G G/H generate, create code 39 extended none on .net projects QR Code Overview TH f (xH )d (xH ) =. G/H H f (xk)dkd (xH ). for all f Cc (G). Hence, for h H , we have f (xh 1 ) (x)dx = G G/H H f (xkh 1 )dkd (xH ). H (h) G/H H H (h) G f (xk)dkd (xH ) f (x) (x)dx. This shows th Code 39 Full ASCII for .NET at (ii) is satis ed. For the last claim, consider Cc (G/H ).

Choose f Cc (G) with TH f = . ( yx) Let y G. Observe that the function x can be viewed as a function (x) on G/H , as a consequence of (i).

Observe also that, if f denotes the function ( yx) x f (yx) , we have (x) (TH f )(xH ) = ( yxh) dh = (xh) H ( yx) = (yxH ) . (x) f (yxh) f (yxh). ( yx) dh (x). Therefore ( yxH ). ( yx) d (xH ) = (x) =. f ( yx). ( yx) (x)dx (x) f (x) (x)dx f ( yx) ( yx)dx = (xH )d (xH ).. This shows the formula of the last claim for the Radon Nikodym derivative. Measures on homogeneous spaces Acontinuous function : G R+ satisfying the identity (i) in Lemma B.1.3 (xh) =. H (h) G (h). (x),. for all x G, h H exists for an y locally compact group G and any closed subgroup H . For the proof, see [Reiter 68, 8, Section 1] or [Folla 95, (2.54)].

Such a function is called a rho-function for the pair (G, H ). Taking this for granted, Part (i) of the following theorem follows from Lemma B.1.

3. Part (iii) is straightforward. For the proof of Part (ii), see [Bou Int2] or [Folla 95, (2.

59)]. Theorem B.1.

4 (i) Quasi-invariant regular Borel measures always exist on G/H . More precisely, given a rho-function for (G, H ), there exists a quasiinvariant regular Borel measure on G/H such that f (x) (x)dx =. G G/H H f (xh)dhd (xH ),. f Cc (G). and with Radon Nikodym derivative dg (gx) (xH ) = , d (x) g, x G. (ii) Any quas bar code 39 for .NET i-invariant regular Borel measure on G/H is associated as above to a rho-function for (G, H ). (iii) If 1 and 2 are quasi-invariant regular Borel measures on G/H , with corresponding rho-functions 1 and 2 , then 1 and 2 are equivalent, with Radon Nikodym derivative d 1 /d 2 = 1 / 2 .

Proposition B.1.5 Let be a quasi-invariant regular Borel measure on G/H as above.

Then the support of is G/H . Proof The proof is similar to the proof of Proposition A.3.

2. Indeed, assume that (U ) = 0 for some open non-empty subset U of G/H . Then (gU ) = 0 for all g G, by quasi-invariance of .

For any compact subset K of G/H , there exists g1 , . . .

, gn G such that K n gi U . Hence, (K) = 0 for i=1 every compact subset K of G/H . Since is regular, this implies that = 0, a contradiction.

Under suitable conditions, there exist relatively invariant measures on homogeneous spaces. A measure on G/H is said to be relatively invariant if, for every g G, there exists a constant (g) > 0 such that g 1 = (g) , that. B.1 Invariant measures dg is a cons barcode code39 for .NET tant, possibly depending on g. d It is clear that is then a continuous homomorphism from G to R+ , called the character of .

Relatively invariant measures are also called semi-invariant measures (see [Raghu 72, page 18]). is, the Radon Nikodym derivative Proposition B.1.

6 (i) Assume that there exists a relatively invariant regular Borel measure on G/H . Then the character of is a continuous extension to G of the homomorphism h H (h)/ G (h) of H . (ii) Assume that the homomorphism H R+ , h H (h)/ G (h) extends .

Then there exists a relatively to a continuous homomorphism : G R+ invariant regular Borel measure with character . Moreover, the measure is unique up to a constant: if is another relatively invariant regular Borel measure on G/H with character , then = c for a constant c > 0. Proof (i) Let be the character of the relatively invariant measure .

The linear functional on Cc (G) given by f . (xH )d (xH ).
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