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Measure Theory Introduction to Fractal Geometry and Chaos

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Measure Theory Introduction to Fractal Geometry and Chaos Measure Theory Introduction to Fractal Geometry and Chaos Matilde Marcolli MAT1845HS Winter 2020, University of Toronto M 1-2 and T 10-12 BA6180 Measure Theory Introduction to Fractal Geometry and Chaos Matilde Marcolli Measure Spaces rigorous theory of \volumes" (and of probabilities) • measure space( X ; Σ; µ) σ-algebra Σ on a set X Σ collection of subsets of X such that ; and X belong to Σ complements: Ac = X r A is in Σ whenever A is in Σ S1 countable unions: k=1 Ak is in Σ if Ak 2 Σ for all k 2 N measure µ :Σ ! R+ [ f1g µ(;) = 0 monotonicity: µ(A1) ≤ µ(A2) if A1 ⊆ A2 1 P1 disjoint additivity: µ(tk=1Ak ) = k=1 µ(Ak ) additive on disjoint unions • probability distribution: a measure space where µ(X ) = 1 A 2 Σ events µ(A) their probabilities Measure Theory Introduction to Fractal Geometry and Chaos Matilde Marcolli Examples X 1 Σ = 2 all subsets of X and µ = δx (Dirac delta measure supported at a chosen point x 2 X ) with 1 x 2 A δ (A) = x 0 x 2= A X 2 Σ = 2 and counting (cardinality) measure #AA finite µ(A) = 1 A infinite 3 If X finite, counting measure can be normalized to a probability (uniform probability) #A µ(A) = #X Measure Theory Introduction to Fractal Geometry and Chaos Matilde Marcolli Support of a measure measure space (X ; Σ; µ) assume Borel measure: (X ; T ) topological space and open sets U 2 T of the topology are all measurable sets U 2 Σ support of µ supp(µ) = fx 2 X j µ(U) > 0 8U 2 T with x 2 Ug an open set U 2 T has µ(U) > 0 iff U \ supp(µ) 6= ; and µ(U) = 0 otherwise Dirac delta measure δx has supp(δx ) = fxg Measure Theory Introduction to Fractal Geometry and Chaos Matilde Marcolli Continuity of measures • limits of sets 1 1 1 1 lim inf An = [k=1 \j=k Aj lim sup An = \k=1 [j=k Aj n!1 n!1 same for increasing or decreasing sequence • (X ; Σ; µ) measure space increasing sequence A1 ⊂ A2 ⊂ · · · An ⊂ · · · µ( lim An) = lim µ(An) n!1 n!1 decreasing sequence A1 ⊃ A2 ⊃ · · · An ⊃ · · · µ( lim An) = lim µ(An) n!1 n!1 arbitrary sequences µ(lim inf An) ≤ lim inf µ(An) n!1 n!1 Measure Theory Introduction to Fractal Geometry and Chaos Matilde Marcolli increasing sequence: limit is union, use 1 1 [n=1An = A1 tk=2 (Ak r Ak−1) and additivity of measure 1 X µ([k Ak ) = µ(A1) + µ(Ak r Ak−1) k=2 1 = lim µ(A1 tk=2 (Ak r Ak−1)) = lim µ(Ak ) k k decreasing sequence: limit is intersection, use Bk = A1 r Ak increasing sequence µ(\k Ak ) = µ(A1) − µ([j Bj ) = µ(A1) − lim µ(Bj ) = lim µ(Aj ) j j 1 arbitrary sequence: use Bk = \j=k Aj increasing sequence 1 µ(lim inf An) = µ([k=1Bk ) = lim µ(Bk ) ≤ lim inf µ(Ak ) n!1 k k Measure Theory Introduction to Fractal Geometry and Chaos Matilde Marcolli Almost everywhere properties • (X ; Σ; µ) measure space, a property of X holds almost everywhere if it fails to hold on a set of µ-measure zero • Examples: X with δx Dirac measure: two functions f1; f2 : X ! R agree almost everywhere iff f1(x) = f2(x) at the point x where δx is supported (sets of measure zero = those not containing x) X with counting measure: two functions f1; f2 : X ! R agree almost everywhere iff f1(x) = f2(x) at all points x 2 X (no non-empty sets of measure zero) • Note: in the case of probability spaces probability zero does not mean impossibility • How to construct more interesting measures than the Dirac and counting examples? Measure Theory Introduction to Fractal Geometry and Chaos Matilde Marcolli Outer measures and measures ∗ X • µ : 2 ! R+ [ f1g defined on all subsets with µ∗(;) = 0 µ(A1) ≤ µ(A2) if A1 ⊆ A2 countable subadditivity: 1 1 ∗ [ X µ ( Ak ) ≤ µ(Ak ) k=1 k=1 even when disjoint union require only ≤ not additivity • use an outer measure µ∗ to construct a σ-algebra Σ and a measure space (X ; Σ; µ) σ-algebra determined by µ∗ ∗ ∗ ∗ Σ := fE ⊆ X : µ (A) = µ (A\E)+µ (A\(X rE)); 8A ⊆ X g ∗ µ = µ jΣ is a measure (the definition of Σ forces additivity on disjoint unions) Measure Theory Introduction to Fractal Geometry and Chaos Matilde Marcolli • σ-algebra property of ∗ ∗ ∗ Σ := fE ⊆ X : µ (A) = µ (A \ E) + µ (A \ (X r E)); 8A ⊆ X g ; and X are in Σ condition defining Σ is symmetric in E and X r E so if a set is in Σ its complement also is given a sequence of sets Ek in Σ ∗ ∗ ∗ µ (A) = µ (A \ E1) + µ (A r E1) = ∗ ∗ ∗ µ (A \ E1) + µ ((A r E1) \ E2) + µ ((A r E1) r E2) = k X ∗ j−1 ∗ k µ ((A r [i=1Ei ) \ Ej ) + µ (A r [j=1Ej ) j=1 k X ∗ j−1 ∗ 1 ≥ µ ((A r [i=1Ei ) \ Ej ) + µ (A r [j=1Ej ) j=1 Measure Theory Introduction to Fractal Geometry and Chaos Matilde Marcolli estimate for all k so in the limit 1 ∗ X ∗ j−1 ∗ 1 µ (A) ≥ µ ((A r [i=1Ei ) \ Ej ) + µ (A r [j=1Ej ) j=1 1 1 j−1 then use A \[j=1Ej = [j=1((A r [i=1Ei ) \ Ej ) ∗ ∗ 1 ∗ 1 µ (A) ≤ µ (A \[j=1Ej ) + µ (A r [j=1Ej ) and by previous also ∗ 1 ∗ 1 ∗ µ (A \[j=1Ej ) + µ (A r [j=1Ej ) ≤ µ (A) 1 so [j=1Ej 2 Σ closed under countable unions Measure Theory Introduction to Fractal Geometry and Chaos Matilde Marcolli ∗ • restriction µ = µ jΣ is a measure disjoint union of sets Ei in Σ, use estimate 1 ∗ X ∗ j−1 ∗ 1 µ (A) ≥ µ ((A r ti=1Ei ) \ Ej ) + µ (A r tj=1Ej ) j=1 applied to A = tj Ej gives 1 ∗ 1 X ∗ µ (tj=1Ej ) ≥ µ (Ej ) j=1 together with subadditivity of the outer measure gives additivity on disjoint unions in Σ Measure Theory Introduction to Fractal Geometry and Chaos Matilde Marcolli Carath´eodory construction • General idea: 1 start with a class of sets S for which have a good definition of what an extension ` (length, area, volume) should be (example: intervals on the real line and their length) 2 define an outer measure on all subsets of the ambient X by setting ∗ X µ (A) := inf `(Sk ) A⊂∪ S k k k infimum over all sequences of sets in the sample class whose union covers A 3 then pass to σ-algebra and measure (X ; Σ; µ) defined by this outer measure 4 this measure µ will still have the property that it agrees with the original ` on sets S 2 S belonging to the chosen sample class, µ(S) = `(S) Measure Theory Introduction to Fractal Geometry and Chaos Matilde Marcolli Existence of non-measurable sets • distinction between the σ-algebra Σ defined by an outer measure and the full set 2X of all subsets of X lies in the possible existence of non-measurable sets • for the case of S = intervals in the real line and ` = length, with µ the Lebesgue measure on R (translation invariant measure) existence of non-measurable sets not constructive: use of the axiom of choice • existence of non-measurable sets on the sphere S2 with respect to the rotation invariant measure related to Banach-Tarski paradox of duplication of the sphere Measure Theory Introduction to Fractal Geometry and Chaos Matilde Marcolli Non-measurable sets: Banach-Tarski paradox F2 free group on two generators a; b all words of arbitrary length in the letters a; b; a−1; b−1 with no cancellations (no a preceded or followed by a−1 and no b preceded or followed by b−1) additional \empty word ; (identity element of the group) product = concatenation of words (with elimination of cancellations) −1 −1 F2 = f;g [ W(a) [W(b) [W(a ) [W(b ) W(x) = words starting with letter x can also write as −1 −1 F2 = W(a) [ a W(a ) or F2 = W(b) [ bW(b ) piece a W(a−1) contains all other possibilities because of cancellation of aa−1, same for bW(b−1) Measure Theory Introduction to Fractal Geometry and Chaos Matilde Marcolli • realize F2 as a subgroup of rigid motions of the sphere choose two different axes of rotation e1; e2 (straight lines through the center of the sphere) choose two angles θ1; θ2 that are not rationally related (θi /π lin independent over Q) group generated by a = Re1,θ1 and b = Re2,θ2 Measure Theory Introduction to Fractal Geometry and Chaos Matilde Marcolli choose a point P on the sphere not on the axes e1; e2 take orbits of P under transformations in each set W(a), W(b), W(a−1), W(b−1) these orbits do not cover the whole sphere S2 keep choosing new points outside of the orbits already obtained and take their orbits under the same sets of transformations until the whole S2 is covered Note: this requires infinitely many choices (axiom of choice is implicit here) Then choose a point xα from each of the orbits constructed (Note: again infinitely many choices) obtain 4 disjoint sets covering the sphere S2 A = [αW(a)xα; B = [αW(b)xα; −1 −1 C = [αW(a )xα; D = [αW(b )xα Measure Theory Introduction to Fractal Geometry and Chaos Matilde Marcolli • duplication of the sphere (Banach-Tarski paradox) break S2 into the pieces A; B; C; D −1 transform C = [αW(a )xα by the rigid motion a = Re1,θ1 0 −1 −1 into C = [αaW(a )xα and transform D = [αW(b )xα by 0 −1 rigid motion b = Re2,θ2 into D = [αbW(b )xα −1 −1 by F2 = W(a) [ aW(a ) = W(b) [ bW(b ) and 2 2 [αF2xα = S obtain two copies of S by reassembling these pieces as 2 0 [ −1 S = A [ C = [αW(a)xα [αaW(a )xα 2 0 [ −1 S = B [ D = [αW(b)xα [αbW(b )xα The decomposition into sets A; B; C; D necessarily involves non-measurable sets! Measure Theory Introduction to Fractal Geometry and Chaos Matilde Marcolli Hausdorff measures µH;s N • ambient space X = R some fixed N with Euclidean distance (or more generally a metric space (X ; d)) ∗ • Hausdorff outer measure for a given s 2 R+ ∗ X s µH;s (A) := lim inf (diamUα) !0 U α where U = fUαg an open covering of A with diameters at most • Hausdorff measure µH;s obtained from the Hausdorff outer measure via the Carath´eodory construction Measure Theory Introduction to Fractal Geometry and Chaos Matilde Marcolli Behavior of the Hausdorff measures P s • for fixed > 0 define µH,,s (E) := infU α(diamUα) for N E ⊂ R , infimum over all countable coverings U = fUαg with 0 < diamUα ≤ N For a given E ⊂ R in the fixed ambient space, the function s 7! µH;s (E) is non-increasing
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