Wavelets in abstract Hilbert space

Size: px

Start display at page:

Download "Wavelets in abstract Hilbert space"

Hester Anthony
6 years ago
Views:

1 Wavelets in abstract Hilbert space Mathieu Sablik Mathematiques, ENS Lyon, 46 allee d Italie, F69364 LYON Cedex 07, France June-July 2000 Introduction The purpose of my training period has been to provide some generalization of the wavelet theory from L 2 to an abstract Hilbert space H. Indeed, a great deal of properties of the wavelet transform can be generalized to abstract Hilbert space based on its fundamental properties alone. In particuar, the aim of this work is to generalize what constitutes the success of wavelets, that is to say to find a vector ψ called the mother wavelet and a group G of unitary operators such that {g.ψ : g G} is a basis of H or better, an orthonormal basis. 1 A first attempt: Multiresolution analysis The first idea is to generalize multiresolution analysis, so we want to find the minimal hypothesis to generalize the multiresolution analysis used for L 2 (R, dµ). Definition 1.1. Let H be a Hilbert space. A multiresolution analysis on H (MRA) consists of a sequence of closed subspaces V j, j Z, a unitary operator γ and a group G of unitary operators satisfying: 1. V j V j+1 for all j Z 2. f V j if and only if γ f V j+1 for all j Z 3. j V j = {0} 4. j V j = H 5. ϕ V 0 such that {g ϕ : g G} is an orthonormal basis for V 0, ϕ is called a scaling function of the given MRA. Work done while visiting Uppsala University, advised by Prof. Kyril Tintarev 1

2 Thanks to (2) and (5) we have by induction that {γ j g ϕ : g G} is an orthonormal basis for V j for all j Z. Then we define by induction closed subspaces W j such that j Z V j+1 = V j W j, so we have H = j W j and to find a mother wavelet of H it suffices only to find a mother wavelet of W 0 associated to G. Proposition 1.2. W j+1 = γ W j j Z Proof. We have: u W j+1 u V j+2 and P Vj+1 u = 0 γ 1 u V j+1 and P Vj γ 1 u = 0 since γ is an unitary γ 1 u W j u γ W j We make now further assumptions, namely that G Z (that is to say that an enumeration of G exists such that for all k, l in Z we have g k g l = g k+l ) and that for all k Z we have g k γ = γ g 2k γ 1 g k = g 2k γ 1. In this case H is separable. All these assumptions are met when H = L 2, γ is the binary dilation operator and g k is the translation by k, i.e. when we have the original construction of orthonormal wavelets developed by Meyer and Mayat. Theorem 1.3. We have γ 1 ϕ V 1 V 0 so (α k ) k Z C Z such that γ 1 ϕ = k α k g k ϕ. In this case, ψ = k Z ( 1)k α 1 k γ g k ϕ is a mother wavelet associated to the group G = {(γ j, g) : j Z, g G}. Proof. First of all, for all l Z we have: g l ψ = k Z( 1) k α k γ g 2l+1 k ϕ To verify the theorem, it suffices to prove that < g k ψ, g l ϕ > for all (k, l) Z 2, span k Z {g k ψ} V 0 = V 1 and < g k, g l >= δ k,l for all (k, l) Z 2. For all j Z we have: < g j ψ, ϕ > = < k ( 1)k α k γ g 2j+1 k ϕ, l α k γ g l ϕ > = (k,l) ( 1)k α k α l < g 2j+1 k ϕ, g l ϕ > = k ( 1)k α k α 2j+1 k = k α 2k α 2j+1 2k k α 2k+1 α 2j 2k = k α 2k α 2j+1 2k k α 2j+1 2k α 2k with k = j k < g j ψ, ϕ > = 0 So for all (k, l) Z 2 we have < g k ψ, g l ϕ >=< g k l ψ, ϕ >= 0. Let W 0 = span k {g k ψ}, we have V 0 W 0 V 1, so we want to show that V 0 W 0 V 1 that is to say γ g k ϕ V 0 W 0 k since {γ g k ϕ : k Z} is a basis of V 1. 2

3 By hypotheses, we can write that g j ϕ = k α k γ g 2j+k ϕ = k α k 2j γ g k ϕ and g j ψ = k ( 1)k α k γ g 2j k+1 ϕ = k ( 1)2j+1 k α 2j+1 k γ g k ϕ. So for some k Z,we deduce: P V0 γ g k ϕ = j α k 2j g j ϕ = (i,j) α k 2jα i 2j γ g i ϕ P W0 γ g k ϕ = j ( 1)2j k+1 α 2j k+1 g j ψ = (i,j) ( 1)i+k α 2j k+1 α 2j+1 i γ g i ϕ So we have: P V0 γ g k ϕ + P W0 γ g k ϕ = i ( α k 2j α i 2j + j j ( 1) i+k α 2j k+1 α 2j+1 i )γ g i ϕ If i + k 2N: j α k 2jα i 2j + j ( 1)i+k α 2j k+1 α 2j+1 i = j α k+2jα i+2j + j α 2j k+1α 2j+1 i = j α k+jα i+j = < g i γ 1 ϕ, g k γ 1 ϕ, > = δ i,k If i + k 2N + 1: j α k 2jα i 2j + j ( 1)i+k α 2j k+1 α 2j+1 i = j α k 2jα i 2j j α 2j k+1α 2j+1 i = j α k 2jα i 2j j α i 2j α k 2j with 2j = 2j + 1 i k = 0 So P V0 γ g k ϕ + P W0 γ g k ϕ = γ g k ϕ for all k Z For all j Z we have: < g j ψ, ψ > = < k ( 1)k α k γ g 2j+1 k ϕ, l ( 1)l α l γ g 1 l ϕ = (k,l) ( 1)k+l α k α l < g 2j+1 k ϕ, g 1 l ϕ > = k ( 1)2k 2j α k α k 2j = (k,l) α l α k 2j < g k ϕ, g l ϕ > = (k,l) α l α k < g l ϕ, g k+2j ϕ > = < l α l γ 1 g l ϕ, k α k γ 1 g k+2j ϕ > = < γ 1 ϕ, g j γ 1 ϕ > < g j ψ, ψ > = δ j,0 So for all (k, l) Z 2 we have < g k ψ, g l ψ >=< g k l ψ, ψ >= δ j,k. This construction uses too specific assumptions, such as g k γ = γ g 2k γ 1 g k = g 2k γ 1 so perhaps it is better to consider another approach: the wavelet transform. 2 Some notions of the group theory This section is to remind, without proof, some results about Haar measure on a locally compact group and about representation of groups on Hilbert spaces which are used in construction of the abstract wavelet transform. 3

4 2.1 The Haar measure Theorem 2.1. Let G be a locally compact group. There is a measure µ on the Borelian σ-algebra, B, called the left-invariant measure such as: 1. µ(k) < if K is compact 2. µ(u) > 0 if U is open 3. µ(xb) = µ(b) for all B B and x G 4. µ(u) = sup{µ(k), K compact, K U}, for all U open 5. µ(b) = inf{µ(u), U open, U B} for all B B Moreover if another measure ν has the same proprieties, then c > 0 such that ν = cµ. Theorem 2.2. L 2 (G, dµ) is a Hilbert space. Theorem 2.3. If T L p (G, dµ), p 1 and U L 1 (G, dµ) then we can define: T U(x) = T (y)u(y 1 x) dµ(y) And we have: T U p T p U 1 G We can also define the convolution product for two functions in L 2 (G, dµ). 2.2 Representations of groups Let H be a Hilbert space and let U(H) be the group of unitary operators acting in H, a homomorphism π from a group G to U(H) is called a unitary representation. Definition 2.4. A representation is called irreducible if and only if the only closed invariant subspaces are the trivial ones {0} and H, or, equivalently if every vector g 0 is cyclic (that is to say that span{π(x)g : x G} = H). Definition 2.5. A unitary representation π of a group G in a Hilbert space H is square integrable if there exists ψ H, ψ 0, such that (x < π(x)ψ, ψ > ) L 2 (G, dµ). ψ is called an admissible vector. 2.3 Schur s lemma and one application Definition 2.6. Let π 1 and π 2 be two representations of the same group G in H 1 and H 2. A bounded operator, A : H 1 H 2, is called an interwining operator iff the following diagram is commutative: H 1 A π 1(x) H 1 A A π 1 (x) = π 2 (x) A H 2 π 2(x) H 2 4

5 Schur s lemma 2.7. Let π 1 and π 2 be two unitary representations of the same group G in H 1 and H 2 such that π 1 (x) is surjective for every x G. Let A : H 1 H 2 be a bounded and interwining operator. So A is a multiple of an isometry. Schur s lemma 2.8. Let A be a bounded operator and let π be an irreducible unitary representation of G on H such that x G π(x) A = A π(x). Then A = λ Id H. Proposition 2.9. Let G be a group, let H and H be two Hilbert spaces, let π be a unitary irreducible representation of G in H, let τ be a unitary representation of G in H and let T be a closed operator from H to H with domain D H dense in H and stable under π. If T π(x) = τ(x)t for every x in G, we have T is a multiple of an isometry and T can be extended to H. Proof. We denote by (.,.) and (.,.) the scalar product in H and H, and by. and. the associated norms. Consider on D the scalar product (g, f) T = (f, g)+(t f, T g) and the assicuated norm. T. Since T is closed, D equipped with the scalar product (.,.) T is a Hilbert space. Since ( T g ) 2 g 2 T = ( T g ) 2 ( T g ) 2 + g 2 1 g H T is bounded from D to H. Then π(x)g 2 T = g 2 +( τ(x) ) 2 = g 2 T for every x G, so π is a unitary representation of G in D. Moreover, for every g G we have g = π(x)π(x 1 )g, so, since D is stable under π(x 1 ), π(x) is surjective for all x G. By the Schur s lemma we have that T is a multiple of an isometry from D to H, so for every g in D we have: ( T g ) 2 = λ g 2 T = λ g 2 + λ( T g ) 2 We deduce that T is a multiple of an isometry from D to H and consequently we can extend T to a multiple of an isometry from H to H. 3 Wavelet transform in abstract Hilbert space Let G be a locally compact group, with left Haar measure dµ (in fact we can do the same thing for the right measure) and π a irreducible, continuous and square integrable unitary representation of G on a Hilbert space H. 3.1 Definition of the wawelet transform Definition 3.1. Let ψ be an admissible vector, let D be the set of vectors f H such that < π(x)ψ, f > L 2 (G, dµ), we can define the wavelet transform: A ψ : D L 2 (G, dµ) f (x < π(x)ψ, f >) We can deduce the following properties: 5

6 Thanks to the Cauchy-Schwarz inegality, we have: x G A ψ f(x) ψ f Since π is square integrable, D, moreover D is invariant under π; as π is irreducible we have D = H. We prove now that A ψ is closed: Let (f n ) n N be a sequence from D converging to f in H and such that A ψ f n converges in L 2 (G, dµ) to ϕ L 2 (G, dµ). By continuity of the scalar product, A ψ f n (x) converges to < π(x)ψ, f > for every x in G, we deduce that < π(.)ψ, f >= ϕ almost everywhere. So < π(x)ψ, f > L 2 (G, dµ) that is to say f D and A ψ f n converges to A ψ f in L 2 (G, dµ). So A ψ is closed. We define for all x in G: τ(x) : L 2 (G, dµ) L 2 (G, dµ) T (y T (x 1 y)) τ is a unitary representation of G in the Hilbert space L 2 (G, dµ). As A ψ π(x)f(y) = < π(y)ψ, π(x)f >=< π(x 1 y)ψ, f >, we have the following commutative diagram: H A ψ π(x) H A ψ A ψ π(x) = τ(x) A ψ L 2 (G, dµ) τ(x) L 2 (G, dµ) We have the conditions of the proposition 2.9, so we deduce the following theorem: Theorem 3.2. A ψ L 2 (G, dµ). can be extanded to a multiple of an isometry from H to Corollary 3.3. If π is square integrable and irreducible, the subset A of all admissible vectors is dense in H. Proof. Indeed, A is a subspace invariant under π, since π is irreducible, we have A = {0} which is impossible or A = H. 3.2 The inversion formula Proposition 3.4. The adjoint operator of A ψ is: A ψ : L2 (G, dµ) H T G T (x) π(x) ψ dµ(x) Remark. In fact this is just a notation which means that this is the vector associated by the Riesz s theorem to the bounded anti-linear functional f < f, T (x) π(x) ψ > dµ(x). G 6

7 Proof. Let f H and T L 2 (G, dµ), we have: < A ψ f, T > L 2 (G,dµ) = < π(x)ψ, f > T (x) dµ(x) G = G < f, π(x)ψ > T (x) dµ(x) = < f, T (x) π(x) ψ dµ(x) > G = < f, A ψ T > Theorem 3.5. Let ψ 1 and ψ 2 be two admissible vectors and let h H be such that h = 1, we define c ψ1,ψ 2 = < A ψ1 h, A ψ2 h > L2 (G,dµ). We have A ψ 2 A ψ1 = c ψ1,ψ 2 Id H. Proof. We have A ψ1 π(x) = τ(x) A ψ1. In the same way, we can show that A ψ 2 τ(x) = π(x) A ψ 2. So we obtain π(x) A ψ 2 A ψ1 = A ψ 2 A ψ1 π(x). By the second version of the Schur s lemma, we deduce that A ψ 2 A ψ1 = λ Id H. The value of the constant is obtained by considering < A ψ1 h, A ψ2 h > L 2 (G,dµ) and in fact it is independent of h. Corollary 3.6. For all f and g in H we have < A ψ1 f, A ψ2 g > = c ψ1,ψ 2 < f, g >. Corollary 3.7. Let ψ 1, ψ 2 and φ in A and f H, we have: A ψ1 f A φ ψ 2 (x) = A ψ1 f(y) A φ ψ 2 (y 1 x) dµ(y) = c ψ1,ψ 2 A φ f(x) 3.3 Study of the image G Proposition 3.8. The image of A ψ1 is exactly the functions T L 2 (G, dµ) such that for all x G T (x) = c 1 ψ1,ψ 2 A ψ1 A ψ 2 T (x). Proof. Let P ψ1,ψ 2 = c 1 ψ1,ψ 2 A ψ1 A ψ 2, we just need to show that P ψ1,ψ2 is a projector. Thanks to the inversion formula, we have: Pψ 2 1,ψ 2 = c 2 ψ1,ψ 2 A ψ1 A ψ 2 A ψ1 A ψ 2 = c 1 ψ1,ψ 2 A ψ1 A ψ 2 = P ψ1,ψ 2 Moreover, A ψ ψ A ψ ψ = A ψ ψ, we deduce that the convolution operator T T A ψ ψ is the orthonormal projection from L 2 (G, dµ) onto Im(A ψ ). So we have another way to obtain the image of A ψ. 3.4 Example In this example we want to show the coherence between the abstract theory and the usual wavelet transform, so here H = L 2 (R). Let G = {(a, b) : a 0, b R} with the group law (a, b) (a, b ) = (a a, b + a b ). G is locally compact and the left Haar measure on G is dµ = a 2 da db. Then we define: π : G U(L 2 (R)) (a, b) (f (f (a,b) : x 1 a f( x b a ))) 7

8 In fact, to prove that π is square integrable and irreducible in view to apply the construction above, it requires a hard analytical argument that repeats much of the classical argument used to prove the same thing. The success of the classical wavelet transform consists in the fact that for some ψ L 2 (R), if we consider the subgroup of G defined by H = {(2 j, k 2 j ) : j Z and k Z} then π(h)ψ is an orthonormal basis. 4 A way of discretisation: the frames As we saw in the exampe above, the wavelet transform contains too much information, that is to say we do not need all the coefficients to recover the function. At the same time it does not seem possible in general case to find a subgroup that provides an orthonormal basis. So we want to find an admissible vector ψ and a subset H G (if possible discrete and better a discrete subgroup) with a new measure dν such that f = H c(f)(h) π(h)ψ dν(h). Of course it is better if c(f) has an easy formula. A first idea is to discretize the formula of A ψ A ψ T, for T L2 (G, dµ), using the Riemann sum, but the convergence is pointwise, so it is dificult to come back in H, and G has to be compact. So we are going to use a method developed by H.G. Feichtinger and K. Gröchenig to construct atomic decomposition in Banach space, related to the integrable group representation in view to built frames in H. 4.1 Generalities about frames Definition 4.1. A collection of elements Ψ = (ψ i ) i I is a frame if there exist constants a and b, 0 < a < b <, such that: a f 2 i I < ψ i, f > 2 b f 2 f H If a = b the frame is called tight, and if a = b = 1 we obtain an orthonormal basis. Then we also define the frame operator: F : H l 2 (I) f {< ψ(i), f >} i Since F is linear and F b, F has an adjoint: F : l 2 (I) H {c i } i I i c i ψ i Let Sf = F Ff = I < ψ i, f > ψ i, for all f H we have a f 2 < f, Sf > b f 2. This shows that S is a positive bounded invertible linear operator and b 1 f 2 < f, S 1 f > a 1 f 2. Lemma 4.2. If Ψ is a frame on H, Ψ = S 1 Ψ is also a frame for H called the dual frame. 8

9 Proof. We have: i < ψ i, f > 2 = i < ψ j, S 1 f > 2 = FS 1 f 2 2 = < (F F)(F F)f, (F F)f > = < f, S 1 f > So the result follows. Theorem 4.3. Let (ψ i ) i I be a frame, we have: f = i I < ψ i, f > ψ i = i I < ψ i, f > ψ i for all f H With convergence in H Proof. Let < f, g > =< f, S 1 g > for f,g in H, this defines an inner product on H equivalent to <.,. > and such that i < ψ i, f > 2 = f 2. So ψ i is an orthonormal basis for <.,. > and we have: f = i < ψ i, f > ψ i = i < ψ i, f > ψ i with convergence in H We also have f = i < ψ i, f > ψ i with weak convergence. To prove the strong convergence, we take J I a finite set and we have: i J < ψ i, f > ψ i = sup g =1 i J < ψ i, f > ψ i, g ( sup g =1 i J < ψ i, f > 2) 1 ( 2 ( sup g =1 g i J < ψi, f > 2) 1 2 i J < ψ i, g > 2) Construction of frames Definition 4.4. A family (x i ) I G I is called U-dense (U G) if i x iu = G. It is called V -separated if for some relatively compact neighbourhood V of the identity, the sets (x i V ) I are pairwise disjoint. And it is called relativelyseparated if it is a finite union of V -separated families. Let N U ((x i ) I ) = sup j I card{i I : x i x j U}, if (x i ) i I is relativelyseparated and U is relatively compact, then N U ((x i ) I ) is finite Definition 4.5. Let U G be a neighborhood of the identity and let (x i ) I be a U-dense and relatively-separated subset. A bounded uniform partition of unity of size U (U-BUPU) is a family Θ = (θ i ) I such that i θ i = 1 almost everywhere, 0 θ i 1 and supp θ i x i U for all i I. Definition 4.6. For U a neighborhood of identity, the U-oscillation of A ψ ψ is: x H Ω U A ψ ψ(x) = sup A ψ ψ(ux) A ψ ψ(x) u U Proposition 4.7. Let T L 2 (G, dµ), (x i ) i I be a relatively-separated family, U be a neighborhood of the identity and Θ be a U-BUPU associated. Then we can define a bounded operator acting on L 2 (G, dµ): T Θ T = i < θ i, T > L 2 (G,dµ) τ xi A ψ ψ with convergence in L 2 (G, dµ) 9

10 Proof. Let J I be a finite set, we have: i J G θ it dµ 2 ( ) 2 i xiu T dµ µ(u) ( ) i J x T iu 2 dµ by Cauchy-Schwarz µ(u) N((x i ) I ) T 2 2 since (x i ) I is relatively separated So (< θ i, T >) i I l 2 (I), since τ xi is an isometry of L 2 (G, dµ), we deduce that T Θ is well defined on L 2 (G, dµ), moreover,we have: T Θ T 2 µ(u) N((x i ) I ) T 2 2 A ψψ 2 So T Θ is bounded and its norm does not depend on Θ. Theorem 4.8. Let (x i ) i I, be a relatively-separated family, U be a neighborhood of the identity, Θ be a U-BUPU associated and ψ A such that A ψ ψ L 1 (G, dµ), Ω U A ψ ψ L 1 (G, dµ), Ω U A ψ ψ 1 < 1 and c ψ,ψ = 1. Then there exists (e i ) i I H I such that: f = i I < e i, f > π(x i )ψ with convergence in H Proof. Let T Im(A ψ ) and x G, since T = T A ψ ψ we have: T (x) T Θ T (x) = [ ( )] G T (y) i θ i(y) A ψ ψ(y 1 x) A ψ ψ(x 1 i x) dµ(y) ( ) G T (y) i θ i(y) sup u U A ψ ψ(y 1 x) A ψ ψ(u y 1 x) dµ(y) So we have: T (x) Ω U A ψ ψ(x) T T Θ T 2 T Ω U A ψ ψ 2 T 2 Ω U A ψ ψ 1 So if F L 2 (G, dµ) A ψ ψ = Im(A ψ ) and if the neighborhood U and ψ are chosen such that Ω U A ψ ψ 1 < 1 then Id T Θ < 1, so T Θ is invertible on Im(A ψ ). Consequently for f H, A ψ f = T Θ T 1 Θ A ψf = ( i < θ i, T 1 Θ A ψf > τ xi A ψ ψ = A ψ i < θ i, T 1 Θ A ψf > π(x i )ψ ). Thanks to the inversion formula, we have: f = i < θ i, T 1 Θ A ψf > π(x i ) ψ For all i I, f < θ i, T 1 Θ A ψf > is a bounded linear functional on H, so e i H such that < θ i, T 1 Θ A ψf >=< e i, f > f H. Now, we have: f = i < e i, f > π(x i ) ψ 10

11 We also have the dual formula f = i < π(x i)ψ, f > e i with weak convergence. Of course we want to have convergence in H, that why we are going to try the same argument as for the frames. So let J I a finite set, we have: i J < π(x i)ψ, f > e i = sup g =1 i J < π(x i)ψ, f > e i, g ( sup g =1 i J < π(xi )ψ, f > 2) 1 ( 2 i J < ei, g > 2) 1 2 ( sup g =1 i J < π(xi )ψ, f > 2) 1 ( 2 i I < θi, T 1 Θ g > 2) 1 2 ( sup g =1 i J < π(xi )ψ, f > 2) 1 2 µ(u)n((xi ) I T Θ 1A ψ g ( µ(u)n((x i ) I T 1 Θ A ψ i J < π(xi )ψ, f > 2) 1 2 Nevertheless, in this argument we must assume that (< π(x i )ψ, f >) i I l 2 (I). In fact, we can always find a such family, but it is difficult to understand within this abstract argument, why in the case of usual wavelets in L 2 (R), the subgroup H = {(2 j, k2 j : j Z and k Z} is enough. Moreover, by this argument we do not prove that we have a frame although we have a recovery of the initial vector. Conclusion Summarily, the wavelet transform makes a connection between an abstract Hilbert space and L 2 (G, dµ). Abstract harmonic analysis provides us powerful tools on L 2 (G, dµ), like the abstract Fourier transform, and makes it is easier to study the abstract Hilbert space. In particular, if G is commutative, we have an extention of the Poisson s formula and when ψ an admissible vector and H is a subgroup of G we obtain a necessary condition for H.ψ being an orthonormal family. This gives a departure point for a more general construction of multiresolution analysis. There are other ways to improve this generalization. Indeed, if we study more the representation theory, we can see that the section 3 can be generalized to Banach space since so can the Schur s lemma. Moreover, in the section 4, we can look for a reconstruction of an element of coorbit of Banach space (If Y H is a Banach space, its coorbit is CoY = {f H : A ψ f Y }). To finish, I conclude by saying that this training period has introduced me into an interesting subject, the abstract harmonic analysis through wavelet transform, which has links with a great deal of other mathematical subjects and which can be used in areas as different as geometry and dynamical systems. Moreover, I thank Kyril Tintarev, the director of my tainning period, for having welcomed me and for the help that he gave me during my trainning period. References [1] E. Hernéndez, G. Weiss: A first course on Wavelets, CRC Press, (1995). [2] Y. Meyer: Ondelettes et operateurs, Hermann, (1990). 11

12 [3] H. G. Feichtinger, K. Gröchenig: Non orthogonal wavelet and Gabor expansions, and group representations, Jones and Bartlett Publishers, (1992). [4] P. Auscher: Wavelet bases for L 2 (R) with rational dilation factor, Jones and Bartlett Publishers, (1992). [5] A. Weil: L integration dans les groupes topologiques, Hermann, (1965). [6] M. Sugiura: Unitary representations and harmonic analysis, Halsted press, (1975) [7] A. Kirillov: Elements of the theory of representation, Springer, Berlin, (1976) 12

MATH MEASURE THEORY AND FOURIER ANALYSIS. Contents

MATH MEASURE THEORY AND FOURIER ANALYSIS. Contents MATH 3969 - MEASURE THEORY AND FOURIER ANALYSIS ANDREW TULLOCH Contents 1. Measure Theory 2 1.1. Properties of Measures 3 1.2. Constructing σ-algebras and measures 3 1.3. Properties of the Lebesgue measure