Pseudo-Additive Measures and Their Applications
Endre Pap, in Handbook of Measure Theory, 2002
DEFINITION 2.18
The pseudo-character of the group [G, +], G n is a continuous [with respect to the usual topology of reals] map ξ: G [a, b], of the group [G, +] into the semiring [[a, b], , ], with property
The map ξ= 0 is trivial pseudo-character.
The forms of pseudo-character in the special cases can be found in Klement et al. [2000a], Pap and Ralevic [1998]. Interesting case for us is [[a, b], , ] = [[0, [, max, +] and then pseudo-character has the form ξ[x, c] = c. x, for each c, where we have taken the dependence of the function ξalso with respect to the parameter c.
Markov Chains
In North-Holland Mathematical Library, 1984
1 Notation
Let N denote the set of positive integers, Z the set of integers and R the set of real numbers; R¯will denote the extended real numbers { } R { } with the usual topology. For a and b in R¯, we write
If E is a set and f a numerical function on E, we shall write
If [E, ] and [F, F] are measurable spaces, we shall write f |F to indicate that the function f : E F is measurable with respect to and F. In the case where F is R and F the σ-algebra of Borel subsets we write simply f . The symbol thus denotes the σ-algebra as well as the set of real-valued measurable functions on E.
If A is a subset of E, we denote the indicator function of A by 1A, so that the statements 1A and A have the same meaning. When A = E we shall often write 1 for 1E.
1.3.If is any set of numerical functions, then + denotes the set of non-negative functions in and b the set of bounded functions. For instance if [E, ] is a measurable space, b is the set of bounded measurable functions and b+ the set of non-negative bounded measurable functions. We recall that b is a Banach space for the supremum norm · described above. We shall denote by U the unit ball of b. The set U+ is thus the set of measurable numerical functions on [E, ] such that for every x E, 0 f[x] 1.
1.4.Let Ω be a set and {[Ei, i]}iI a collection of measurable sets. For each i I let fi be a map from Ω to Ei; the σ-algebra on Ω generated by the sets {fi1[Ai]: Ai Ei} is the smallest σ-algebra on Ω relative to which all the fi are measurable, and it will be denoted by σ[fi, i I]. If F is any collection of subsets of E, we denote by σ[F] the σ-algebra generated by F. A measurable space [E, ] will be said to be separable if is generated by a countable collection of sets.
1.5.We denote by M[] the space of σ-finite measures on [E, ] and by bM[] the space of finite measures. We shall call μ+ and μ the positive and negative parts of the measure μ and set |μ| = μ+ μ. The space bM[] is a Banach space when endowed with the norm μ = |μ| [E].
If f and the integral of f with respect to the measure μ exists, we shall write it in any of the forms
The space of integrable functions will be denoted by 1[μ], and as usual L1[μ] will be the Banach space of equivalence classes of integrable functions endowed with the norm f1 = E |f| dμ. There is a canonical isometry of LI[μ] into bM[] which maps f onto the measure λ : A A f dμ; λ is absolutely continuous with respect to μ [λ μ].
1.6.If [Ω, F, P] is a probability space we shall, as is customary, use the words random variable and expectation in place of measurable function and integral, and we shall write
If G is a sub-σ-algebra of , we denote E[X|G] the conditional expectation of X [assumed to be quasi-integrable] given G. When X = 1A, A , we write P[A | G] instead of E[1A | G], and if G is the smallest σ-algebra relative to which the random variable Y is measurable, we may write E[X | Y] and P[A | Y]. When we apply conditional expectation successively, we shall often abbreviate E[E[X | G1] | G2] by E[X | G1 | G2]. We recall that conditional expectations are defined up to P-equivalence; in the sequel we shall often omit the qualifying phrase almost everywhere relative to P in the relations involving conditional expectations.
Basic Representation Theory of Groups and Algebras
In Pure and Applied Mathematics, 1988
1.18
Here are a few simple examples of topological groups.
Any [untopologized] group becomes a topological group when equipped with the discrete topology. Perhaps the most important infinite discrete group is the additive group of the integers [the infinite cyclic group].
The real number field , with its usual topology and the operation of addition, forms a second-countable connected locally compact group called the additive group of the reals. The set of all non-zero real numbers, with the relativized topology of and the operation of multiplication, forms a second-countable locally compact group * called the multiplicative group of non-zero reals. This group is not connected; its connected component of the unit is the multiplicative subgroup ++ of all positive real numbers.
The same of course can be said of the complex field , except that the multiplicative group = \ {0} is connected.
Let us denote {z :|z| = 1} by E. With the operation of multiplication, and with the relativized topology of the complex plane, Ebecomes a second-countable connected compact group; it is called the circle group. It plays an enormous role in the theory of unitary group representations.
Half-Linear Differential Equations
In North-Holland Mathematics Studies, 2005
8.3.1 Essentials on time scales, basic properties
In 1988, Stefan Hilger [178] introduced the calculus on time scales in order to unify continuous and discrete analysis. By a time scale T [an alternative terminology is measure chain] we understand any closed subset of the real numbers with the usual topology inherited from . Typical examples of time scales are T = and T = the set of integers [or h denned as {hk : k } with a positive h]. The operators ρ, σ : T T are defined by
A function f : T is called rd-continuous provided it is continuous at all right-dense points in T and its left-sided limits exist [finite] at all left-dense points in T. For a, b T and a delta differentiable function f, the Newton integral is defined by ƒba fΔ[t] Δt = f[b] f[a]. For the concept of the Riemann delta integral and the Lebesgue delta integral see [50, Chapter 5]. Note that we have
while
These are the most typical time scales, but there exist much more examples, which may bring quite surprising unusual [and unpleasant, sometimes] phenomena in some aspects of the theory. Let us mention at least T = q0 = {qk : k; 0} [or T = q{0}], where q > 1 is a real number. Then σ[t]=qt, and a dynamic equation considered on such a time scale is called q-difference equation. The last example of the time scale which we present here is the set defined as the union of closed mutually disjoint intervals. The basic facts of the time scale calculus can be found in [178] and the general theory of dynamic equations on time scales along with an excellent introduction into the subject is presented in [49, 50].
Now consider the linear dynamic equation on a time scale T
where xσ=xºσand r, c : T with r[t] 0, and its half-linear extension
Obviously, [8.3.2] reduces to [1.1.1] if T = and to [8.1.4] if T = , respectively.
By means of the approach, which is an extension of that in Subsection 1.1.6, it can be shown that the initial value problem involving equation [8.3.2] is globally uniquely solvable provided the coefficients r, c are rd-continuous. However, it has to mentioned, that the part with a reciprocal equation cannot be extended reasonably here, since it requires Δ and σto be commutative [i.e., fΔσ=fσΔ], which is not true in general. Also the approach based on the Prüfer transformation has not been developed yet. The reason is that there is no real chain rule for differentiation on time scales.
Real Reductive Groups I
In Pure and Applied Mathematics, 1988
Theorem
Sa,b[G] is a Fréchet space.
[1]The inclusion of Cc[G]into Sa,b[G] is continuous with dense image. [Here Cc[G]is given the usual topology, which will be described in the course of the proof.]
[2]If f Sa,b[G] and if x, y U[g], r 0 then h[g] = a[g]rL[x]R[y]f[g] is in L2[G]. Furthermore, the map
If γ K and if f Sa,b[G] then Eyf Sa,b[G] and the series Σ Eγf converges to f in Sa,b[G].
Since we may use the pa,b,x,y,r with x, y running through a basis of U[g] and r rational, to prove that S = Sa,b[G] is a Fréchet space it is enough to show that it is sequentially complete. Let fj be a Cauchy sequence in S.
The definition of the topology of S easily implies that there exists f C[G] such that fj converges uniformly with all derivatives to f on compacta [c.f. the argument in 1.6.4]. Let Ω be an open subset of G with compact closure. We set for h C[G],
Then pa,b,x,y,d[h] = supΩ Ωpx,y,d[h].
Fix x, y, d. Let N be so large that [px,y,d = pa,b,x,y,d] if j, k N then Px,y,d[fj fk] < 1. Then px,y,d[fj] < 1 + px,y,d[fN] for j N. Let Ω be open in G with compact closure and let NΩ N be such that Ωpx,y,d[f fj] < 1 for j NΩ. Then Ωpx,y,d[f] < 2 + Ωpx,y,d[fN]. Hence,
Let h C[R] be such that h[x] = 1 if |x| 1 and h[x] = 0 if |x| 2. Set for r > 0, ur[g] = 1[a[g]/r]. We leave it to the reader to check
[i]Let x, y U[g] then there exists a constants Bx,y and D[x, y] such that |L[x]R[y]ur[g]| Bx,ya[g]D[x,y] for r > 1 and all g G.
Let x, y Uk[g] and let x1,, xq be a basis of Uk[g]. Then
for r > 1. Thus urf f in S. We have thus proved [1]
The first part of [2] follows from [3] above. Let f1, f2 S. Then
The absolute value of the right hand side is at most
times
Now [4] implies that a[z1 g] a[z]1 a[g]. Hence
Thus if we take d2 = d and d1 > d + d0 then
We now prove the last assertion. Let j be fixed and let x1,, xn be a basis of Uj[g]. If x Uj[g] then Ad[k]x = Σ uj[k]xj with uj a matrix coefficient of [Ad|K, Uj [g]]. Fix T a maximal torus of K and Φ+ a system of positive roots for Φ[tC,tC].. Let Σ denote the set of weights of T on Uj[g]. Let ρk be [as usual] the half sum of the elements of Φ+. If γ K then set μγ equal to the highest weight of γ. Let CK be the Casimir operator of frelative to B|t.. Then the eigenvalue if CK on any representative of γ is μγ + ρk2 ρk2.
[ii]Set D = ρk2 + CK. Then DrEγf = EγDrf. So
[*]Let Sj[γ] denote the set elements of K that occur in Vγ Uj[g]. If σ Sj[γ] then μσ = μγ + σ with δ a weight of the action of T on Uj[gC]. This we have
[iii]If σ Sj[γ] then μσ + pK Cjμγ + ρk + Dj
with Cj and Dj positive constants depending only on j.
Let f S. Then a[g]r b[g]1|L[x]R[y]Eγf[g]| =
Now, |χγ[k]| d[γ] and |uj[k]| Cx for k K. We therefore conclude that
[iv]In the integral above we may replace L[xj]R[y]f[k 1 g] by
If we apply [iv] and [*] above we have
[v]The Weyl dimension formula implies that d[γ] Cμγ + ρkm with m = |Φ+|. Thus if μγ + ρk > Dj/2Cj then
[vi]This combined with [ii] easily implies that ΣγEγf converges in S. The argument in 1.4.7 easily implies that the above series converges pointwise to f.
This completes the proof of the theorem.
Basic Representation Theory of Groups and Algebras
In Pure and Applied Mathematics, 1988
Measurability of Complex Functions
1.8
Suppose that X and Y are two sets, and that L and are σ-fields of subsets of X and Y respectively. A function f: X Y is called an L, Borel function if f1[A] L whenever A . It is of course enough to require that f1[A] L for all A in some subfamily of which generates .
Suppose that Y = and that is generated by the usual topology of . Then if f and g are L, Borel functions on X to , the functions |f|, cf[c ], f + g, and fg are also L, Borel functions.
Next, assume that Y = ext and that is generated by the usual topology of ext. If f, g, f1, f2, f3, are any L, Borel functions on X to ext, then the functions
are all L, Borel.
These facts are proved in any standard text on Lebesgue integration.
1.9
Now let μ be a complex measure on a δ-ring L of subsets of a set X. A function f on X to [or to ext] is said to be locally μ-measurable if it is L, Borel, where L is the σ-field of locally μ-measurable subsets of X, and is the σ-field generated by the usual topology of [or of ext].
Thus the results of 1.8 apply to locally μ-measurable functions.
If two complex [or ext-valued] functions on X coincide locally μ-almost everywhere, and one is locally μ-measurable, then so is the other.
If X L, or if we are talking about the behavior of a function f only on a fixed set A in L, we will omit the word locally and speak of f as μ-measurable instead of locally μ-measurable.
1.10
Suppose f is a complex function on X such that, for each A in L, there is a sequence {gn} of locally μ-measurable complex functions on X satisfying limngn[x] = f[x] for μ-almost all x in A. Then evidently f is locally μ-measurable.
1.11 Egoroff's Theorem
Let {fn} be a sequence of locally μ-measurable complex-valued functions on X, and f a complex-valued function on X such that limn fn[x] = f[x] for locally μ-almost all x. Then for any A in L and any positive number ε, there is a set B in L such that B A, |μ|[A \ B]< ε, and fn[x] f[x] uniformly for x in B.
For the proof see Halmos [1], Theorem 21A.
Real Reductive Groups I
In Pure and Applied Mathematics, 1988
5.2.7
Fix a subset, F, of Δ0. Let [σ, Hσ] be an admissible, finitely generated, Hilbert representation of 0MF. Let μ[aF]C*. Let [πPF,σ,μ,HP,σ,μ]be as above. Since PF will be fixed in this number, we will drop the PF in our notation. Let [Hσ] be endowed with the usual topology [1.6.3]. We set [Hσ,μ] equal to the space of a smooth functions from G to [Hσ] that are in Hσ,μ. If x U[g] and if δ is one of the semi-norms defining the topology on [Hσ,μ] we set δx[f] = supk K δ[πσ,μ[x]f[k]] for f [Hσ,μ]. Then it is easy to see that [Hσ,μ] defines a smooth Fréchet representation of G, that IPF,σ,μis a dense subspace and that [Hσ,μ] is contained in [Hσ,μ].
It can be shown [c.f. Borel, Wallach [1, III, 7.9] that [Hσ,μ] is equivalent to [Hσ,μ] as a smooth Fréchet module.
Sobolev Spaces
In Pure and Applied Mathematics, 2003
Topological Vector Spaces
1.4 Topological Spaces
If X is any set, a topology on X is a collection O of subsets of X which contains
[i]the whole set X and the empty set ø,
[ii]the union of any collection of its elements, and
[iii]the intersection of any finite collection of its elements.
The pair [X, O] is called a topological space and the elements of O are the open sets of that space. An open set containing a point x in X is called a neighbourhood of x. The complement X U = {x X : x U} of any open set U is called a closed set. The closure S¯of any subset S X is the smallest closed subset of X that contains S.
Let O1 and O2 be two topologies on the same set X. If O1 O2, we say that O2 is stronger than O1, or that O1 is weaker than O2.
A topological space [X, O] is called a Hausdorff space if every pair of distinct points x and y in X have disjoint neighbourhoods.
The topological product of two topological spaces [X, OX] and [Y, OY] is the topological space [X × Y, O], where X × Y = {[x, y] : x X, y Y} is the Cartesian product of the sets X and Y, and O consists of arbitrary unions of sets of the form {OX × OY : OX OX, OY OY}.
Let [X, OX] and [Y, OY] be two topological spaces. A function f from X into Y is said to be continuous if the preimage f1[O] = {x X : f [x] O} belongs to OX for every O OY. Evidently the stronger the topology on X or the weaker the topology on Y, the more such continuous functions f there will be.
1.5 Topological Vector Spaces
We assume throughout this monograph that all vectors spaces referred to are taken over the complex field unless the contrary is explicitly stated.
A topological vector space, hereafter abbreviated TVS, is a Hausdorff topological space that is also a vector space for which the vector space operations of addition and scalar multiplication are continuous. That is, if X is a TVS, then the mappings
from the topological product spaces X × X and × X, respectively, into X are continuous. [Here has its usual topology induced by the Euclidean metric.]
X is a locally convex TVS if each neighbourhood of the origin in X contains a convex neighbourhood of the origin.
We outline below those aspects of the theory of topologicaland normed vector spaces that play a significant role in the studyof Sobolev spaces. For a more thorough discussion of these topics the reader is referred to standard textbooks on functional analysis, for example [Ru1] or [Y].
1.6 Functionals
A scalar-valued function defined on a vector space X is called a functional. The functional f is linear provided
If X is a TVS, a functional on X is continuous if it is continuous from X into where has its usual topology induced by the Euclidean metric.
The set of all continuous, linear functionals on a TVS X is called the dual of X and is denoted by X. Under pointwise addition and scalar multiplication X is itself a vector space:
X will be a TVS provided a suitable topology is specified for it. One such topology is the weak-star topology, the weakest topology with respect to which the functional Fx, defined on X by Fx[f] = f[x] for each f X, is continuous for each x X. This topology is used, for instance, in the space of Schwartz distributions introduced in Paragraph 1.57. The dual of a normed vector space can be given a stronger topology with respect to which it is itself a normed space. [See Paragraph 1.11.]
Preliminaries of Dynamical Systems Theory
H.W. Broer, F. Takens, in Handbook of Dynamical Systems, 2010
1.3.6 Shifts
A very different class of dynamical systems is formed by the shiftshere one also speaks of symbolic dynamics. In this case the state space is no longer a manifold. In the simplest case it is the space of all sequences S=[si]whose elements belong to a given finite set A, called the alphabet, and whose indices have values in Zor in Z+depending on whether the time set is Zor Z+. The usual topology on such a state spaces can be given by the metric ρwhich assigns to two sequences S=[si]and T=[ti]a distance 2kif kis the smallest absolute value of an index ifor which the corresponding elements siand tiare different. This topology on this space of sequences, which can be identified with the infinite product AZor AZ+, is the so called product topology [where Ais assumed to have the discrete topology]. The evolution map Φassigns to the sequence S=[si]and nTthe sequence Φ[S,n]=σn[S]=S~=[s~i=si+n]. One calls σthe shift map. These spaces of sequences are also called shift spaces.
A systems whose state space is a closed subset of AZor AZ+which is invariant under the shift map is called asub-shift.
These dynamical systems are important as a link between probability theory and differentiable dynamics: [sub-]shifts, with invariant measures on the state space, are at the basis of Bernoulli and Markov processes. They are even at the basis of the general invariant measures constructed by thermodynamic formalism. On the other hand one can prove that many differentiable dynamical systems on manifolds have invariant sets [so called horseshoes, and more general so-called hyperbolic basic sets] which are homeomorphic, as a space with an evolution map, to such a shift. A classical reference for these shifts and their relation to smooth dynamics is[10]; for more information see also[34].