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We study metric spheres (Z,dZ) obtained by gluing two hemispheres of S2 along an orientation-preserving homeomorphism g:S1→S1, where dZ is the canonical distance that is locally isometric to S2 off the seam. We show that if (Z,dZ) is quasiconformally equivalent to S2, in the geometric sense, then g is a welding homeomorphism with conformally removable welding curves. We also show that g is bi-Lipschitz if and only if (Z,dZ) has a 1-quasiconformal parametrization whose Jacobian is comparable to the Jacobian of a quasiconformal mapping h:S2→S2. Furthermore, we show that if g−1 is absolutely continuous and g admits a homeomorphic extension with exponentially integrable distortion, then (Z,dZ) is quasiconformally equivalent to S2. peerReviewed

Let $A,B \subset \mathbb{R}$ be closed Ahlfors-regular sets with dimensions $\dim_{\mathrm{H}} A =: \alpha$ and $\dim_{\mathrm{H}} B =: \beta$. I prove that $$\dim_{\mathrm{H}} [A + \theta B] \geq \alpha + \beta \cdot \tfrac{1 - \alpha}{2 - \alpha}$$ for all $\theta \in \mathbb{R} \, \setminus \, E$, where $\dim_{\mathrm{H}} E = 0$., Comment: 44 pages. v2: incorporated referee comments, to appear in GAFA

We consider unions of $SL(2)$ lines in $\mathbb{R}^{3}$. These are lines of the form $$L = (a,b,0) + \mathrm{span}(c,d,1),$$ where $ad - bc = 1$. We show that if $\mathcal{L}$ is a Kakeya set of $SL(2)$ lines, then the union $\cup \mathcal{L}$ has Hausdorff dimension $3$. This answers a question of Wang and Zahl. The $SL(2)$ lines can be identified with horizontal lines in the first Heisenberg group, and we obtain the main result as a corollary of a more general statement concerning unions of horizontal lines. This statement is established via a point-line duality principle between horizontal and conical lines in $\mathbb{R}^{3}$, combined with recent work on restricted families of projections to planes, due to Gan, Guo, Guth, Harris, Maldague, and Wang. Our result also has a corollary for Nikodym sets associated with horizontal lines, which answers a special case of a question of Kim., Comment: 7 pages

We extend the classical Carath\'eodory extension theorem to quasiconformal Jordan domains $( Y, d_{Y} )$. We say that a metric space $( Y, d_{Y} )$ is a quasiconformal Jordan domain if the completion $\overline{Y}$ of $( Y, d_{Y} )$ has finite Hausdorff $2$-measure, the boundary $\partial Y = \overline{Y} \setminus Y$ is homeomorphic to $\mathbb{S}^{1}$, and there exists a homeomorphism $\phi \colon \mathbb{D} \rightarrow ( Y, d_{Y} )$ that is quasiconformal in the geometric sense. We show that $\phi$ has a continuous, monotone, and surjective extension $\Phi \colon \overline{ \mathbb{D} } \rightarrow \overline{ Y }$. This result is best possible in this generality. In addition, we find a necessary and sufficient condition for $\Phi$ to be a quasiconformal homeomorphism. We provide sufficient conditions for the restriction of $\Phi$ to $\mathbb{S}^{1}$ being a quasisymmetry and to $\partial Y$ being bi-Lipschitz equivalent to a quasicircle in the plane., Comment: 21 pages; revised version

We study the modulus of dual families of $k$- and $(n-k)$-dimensional Lipschitz chains of Euclidean $n$-cubes and establish half of the modulus duality identity.

I prove that a closed $n$-regular set $E \subset \mathbb{R}^{d}$ with plenty of big projections has big pieces of Lipschitz graphs. This answers a question of David and Semmes., Comment: 40 pages. v3: referee comments incorporated

We generalize a result of Freedman and He [4, Theorem 2.5], concerning the duality of moduli and capacities in solid tori, to sufficiently regular metric spaces. This is a continuation of the work of the author and Rajala [12] on the corresponding duality in condensers. peerReviewed

We prove that there exists a constant $\epsilon> 0$ with the following property: if $K \subset {\mathbb {R}}^2$ is a compact set that contains no pair of the form $\{x, x + (z, z^{2})\}$ for $z \neq 0$, then $\dim _{\textrm {H}} K \leq 2 - \epsilon $.

A Semmes surface in the Heisenberg group is a closed set $S$ that is upper Ahlfors-regular with codimension one and satisfies the following condition, referred to as Condition B. Every ball $B(x,r)$ with $x \in S$ and $0 < r < \operatorname{diam} S$ contains two balls with radii comparable to $r$ which are contained in different connected components of the complement of $S$. Analogous sets in Euclidean spaces were introduced by Semmes in the late $80$'s. We prove that Semmes surfaces in the Heisenberg group are lower Ahlfors-regular with codimension one and have big pieces of intrinsic Lipschitz graphs. In particular, our result applies to the boundary of chord-arc domains and of reduced isoperimetric sets., 39 pages, 4 figures

This note concerns low-dimensional intrinsic Lipschitz graphs, in the sense of Franchi, Serapioni, and Serra Cassano, in the Heisenberg group $\mathbb{H}^n$, $n\in \mathbb{N}$. For $1\leq k\leq n$, we show that every intrinsic $L$-Lipschitz graph over a subset of a $k$-dimensional horizontal subgroup $\mathbb{V}$ of $\mathbb{H}^n$ can be extended to an intrinsic $L'$-Lipschitz graph over the entire subgroup $\mathbb{V}$, where $L'$ depends only on $L$, $k$, and $n$. We further prove that $1$-dimensional intrinsic $1$-Lipschitz graphs in $\mathbb{H}^n$, $n\in \mathbb{N}$, admit corona decompositions by intrinsic Lipschitz graphs with smaller Lipschitz constants. This complements results that were known previously only in the first Heisenberg group $\mathbb{H}^1$. The main difference to this case arises from the fact that for $1\leq k, Comment: 30 pages; v2: minor revision, results unchanged

In this paper we discuss the convergence of distances associated to converging structures of Lipschitz vector fields and continuously varying norms on a smooth manifold. We prove that, under a mild controllability assumption on the limit vector-fields structure, the distances associated to equi-Lipschitz vector-fields structures that converge uniformly on compact subsets, and to norms that converge uniformly on compact subsets, converge locally uniformly to the limit Carnot-Carathéodory distance. In the case in which the limit distance is boundedly compact, we show that the convergence of the distances is uniform on compact sets. We show an example in which the limit distance is not boundedly compact and the convergence is not uniform on compact sets. We discuss several examples in which our convergence result can be applied. Among them, we prove a subFinsler Mitchell’s Theorem with continuously varying norms, and a general convergence result for Carnot-Carathéodory distances associated to subspaces and norms on the Lie algebra of a connected Lie group.

Tässä tutkielmassa todistetaan Marstrandin projektiolause käyttäen apuna potentiaaliteoriaa. Projektiolauseen mukaan 2-ulotteisen Borel joukon ortogonaaliprojektion Hausdorffin dimensio on luvun 1 ja kyseisen Borel joukon dimension minimi melkein kaikkiin eri suuntiin. Intuitiivisesti lause kertoo, että joukon varjon dimensio on suurin mahdollinen. Marstrandin projektiolauseen todistamiseksi tutkielmassa rakennetaan teoria alkaen yleisen mittateorian perustuloksista. Mittateorian pohjalta määritellään Hausdorffin mitta, jonka avulla määritellään joukon Hausdorffin dimensio. Intuitiivisesti Hausdorffin dimensio kuvaa joukon geometrista kokoa ja se on yksikäsitteinen jokaiselle joukolle. Hausdorffin dimensio mahdollistaa monimutkaisten joukkojen, kuten fraktaalien, geometrian tutkimisen. Lisäksi esitellään dimensioihin liittyviä merkintöjä ja tapoja arvioida joukon Hausdorffin dimension suuruutta. Tutkielman lopussa esitellään algoritminen menetelmä, jonka avulla voidaan muodostaa esimerkkejä fraktaaleista. Lopuksi sovelletaan Marstrandin projektiolausetta erilaisiin joukkoihin. John Marstrand todisti projektiolauseen vuonna 1954. Robert Kaufman todsti tuloksen käyttäen potentiaaliteoriaa vuonna 1968. Myöhemmin Kenneth Falconer esitteli Kaufmania mukaillen potentiaaliteoriaan perustuvan todistuksen. Tässä tutkielmassa esitellään kyseinen todistus yksityiskohtaisemmin. Marstrandin projektiolause tuli tunnetuksi, kun Mandelbrot popularisoi fraktaalin käsitteen 1970-luvulla. Lause voidaan yleistää korkeampiin dimensioihin ja se on tärkeä työkalu fraktaalien geometrian tarkastelussa. Vaikka lause on tunnettu pitkään, siihen liittyy edelleen avoimia ongelmia. In this thesis we prove Marstrand's projection theorem using potential theoretical methods. Projection theorem claims that the Hausdorff dimension of the orthogonal projection of a Borel set in $\mathbb{R}^2$ is the minimum between 1 and the dimension of the set for almost all angles. Intuitively, the theorem gives that the shadow of the set has the highest possible dimension. This result was first proven by John Marstrand in 1954 and it became well known after Mandelbrot popularized the notion of fractal in the 1970s. Marstrand’s theorem has generalizations to higher dimensions and it is an important tool to look into the geometry of fractals. Although the theorem has been known for long time, there are still open problems related to it.

Two definitions for the rectifiability of hypersurfaces in Heisenberg groups Hn have been proposed: one based on H-regular surfaces and the other on Lipschitz images of subsets of codimension-1 vertical subgroups. The equivalence between these notions remains an open problem. Recent partial results are due to Cole–Pauls, Bigolin–Vittone, and Antonelli–Le Donne. This paper makes progress in one direction: the metric Lipschitz rectifiability of H-regular surfaces. We prove that H-regular surfaces in Hn with α-Hölder continuous horizontal normal, α>0⁠, are metric bilipschitz rectifiable. This improves on the work by Antonelli–Le Donne, where the same conclusion was obtained for C∞-surfaces. In H1⁠, we prove a slightly stronger result: every codimension-1 intrinsic Lipschitz graph with an ϵ of extra regularity in the vertical direction is metric bilipschitz rectifiable. All the proofs in the paper are based on a new general criterion for finding bilipschitz maps between “big pieces” of metric spaces. peerReviewed

We give a geometric criterion for a topological surface in the first Heisenberg group to be an intrinsic Lipschitz graph, using planar cones instead of the usual open cones. peerReviewed

We give a characterization of metric spaces quasisymmetrically equivalent to a finitely connected circle domain. This result generalizes the uniformization of Ahlfors 2-regular spaces by Merenkov and Wildrick. peerReviewed

We study properties of functions with bounded variation in Carnot-Caratheodory spaces. We prove their almost everywhere approximate differentiability and we examine their approximate discontinuity set and the decomposition of their distributional derivatives. Under an additional assumption on the space, called property R , we show that almost all approximate discontinuities are of jump type and we study a representation formula for the jump part of the derivative.

We study critical regularity assumptions on space-filling curves that possess certain modulus of continuity. The bounds we obtain are essentially sharp, as demonstrated by an example. peerReviewed

The purpose of this paper is to introduce and study some basic concepts of quantitative rectifiability in the first Heisenberg group $\mathbb{H}$. In particular, we aim to demonstrate that new phenomena arise compared to the Euclidean theory, founded by G. David and S. Semmes in the 90's. The theory in $\mathbb{H}$ has an apparent connection to certain nonlinear PDEs, which do not play a role with similar questions in $\mathbb{R}^{3}$. Our main object of study are the intrinsic Lipschitz graphs in $\mathbb{H}$, introduced by B. Franchi, R. Serapioni and F. Serra Cassano in 2006. We claim that these $3$-dimensional sets in $\mathbb{H}$, if any, deserve to be called quantitatively $3$-rectifiable. Our main result is that the intrinsic Lipschitz graphs satisfy a weak geometric lemma with respect to vertical $\beta$-numbers. Conversely, extending a result of David and Semmes from $\mathbb{R}^{n}$, we prove that a $3$-Ahlfors-David regular subset in $\mathbb{H}$, which satisfies the weak geometric lemma and has big vertical projections, necessarily has big pieces of intrinsic Lipschitz graphs., Comment: 56 pages, one figure. v3: incorporated referee suggestions, to appear in Amer. J. Math

This note concerns Loomis-Whitney inequalities in Heisenberg groups $\mathbb{H}^n$: $$|K| \lesssim \prod_{j=1}^{2n}|\pi_j(K)|^{\frac{n+1}{n(2n+1)}}, \qquad K \subset \mathbb{H}^n.$$ Here $\pi_{j}$, $j=1,\ldots,2n$, are the vertical Heisenberg projections to the hyperplanes $\{x_j=0\}$, respectively, and $|\cdot|$ refers to a natural Haar measure on either $\mathbb{H}^n$, or one of the hyperplanes. The Loomis-Whitney inequality in the first Heisenberg group $\mathbb{H}^1$ is a direct consequence of known $L^p$ improving properties of the standard Radon transform in $\mathbb{R}^2$. In this note, we show how the Loomis-Whitney inequalities in higher dimensional Heisenberg groups can be deduced by an elementary inductive argument from the inequality in $\mathbb{H}^1$. The same approach, combined with multilinear interpolation, also yields the following strong type bound: $$\int_{\mathbb{H}^n} \prod_{j=1}^{2n} f_j(\pi_j(p))\;dp\lesssim \prod_{j=1}^{2n} \|f_j\|_{\frac{n(2n+1)}{n+1}}$$ for all nonnegative measurable functions $f_1,\ldots,f_{2n}$ on $\mathbb{R}^{2n}$. These inequalities and their geometric corollaries are thus ultimately based on planar geometry. Among the applications of Loomis-Whitney inequalities in $\mathbb{H}^n$, we mention the following sharper version of the classical geometric Sobolev inequality in $\mathbb{H}^n$: $$\|u\|_{\frac{2n+2}{2n+1}} \lesssim \prod_{j=1}^{2n}\|X_ju\|^{\frac{1}{2n}}, \qquad u \in BV(\mathbb{H}^n),$$ where $X_j$, $j=1,\ldots,2n$, are the standard horizontal vector fields in $\mathbb{H}^n$., Comment: 27 pages. This paper supersedes arXiv:2003.05862v1; Loomis-Whitney inequalities are obtained in higher-dimensional Heisenberg groups by a simpler approach. Some preliminaries and Section 4 draw heavily from arXiv:2003.05862v1, where they were stated for the first Heisenberg group

We consider applications of the dual pair of the (upper) Assouad dimension and the lower (Assouad) dimension in analysis. We relate these notions to other dimensional conditions such as a Hausdorff content density condition and an integrability condition for the distance function. The latter condition leads to a characterization of the Muckenhoupt Ap properties of distance functions in terms of the (upper) Assouad dimension. It is also possible to give natural formulations for the validity of Hardy–Sobolev inequalities using these dual Assouad dimensions, and this helps to understand the previously observed dual nature of certain cases of these inequalities. peerReviewed

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In this paper we provide a proof of the Carleson 𝜀2-conjecture. This result yields a characterization (up to exceptional sets of zero length) of the tangent points of a Jordan curve in terms of the finiteness of the associated Carleson 𝜀2-square function. peerReviewed

This paper contributes to the study of sets of finite intrinsic perimeter in Carnot groups. Our intent is to characterize in which groups the only sets with constant intrinsic normal are the vertical half-spaces. Our viewpoint is algebraic: such a phenomenon happens if and only if the semigroup generated by each horizontal half-space is a vertical half-space. We call semigenerated those Carnot groups with this property. For Carnot groups of nilpotency step 3 we provide a complete characterization of semigeneration in terms of whether such groups do not have any Engel-type quotients. Engel-type groups, which are introduced here, are the minimal (in terms of quotients) counterexamples. In addition, we give some sufficient criteria for semigeneration of Carnot groups of arbitrary step. For doing this, we define a new class of Carnot groups, which we call type (◊)(◊) and which generalizes the previous notion of type (⋆)(⋆) defined by M. Marchi. As an application, we get that in type (◊)(◊) groups and in step 3 groups that do not have any Engel-type algebra as a quotient, one achieves a strong rectifiability result for sets of finite perimeter in the sense of Franchi, Serapioni, and Serra-Cassano. peerReviewed

We formalize the notion of limit of an inverse system of metric spaces with 1-Lipschitz projections having unbounded fibers. The construction is applied to the sequence of free Carnot groups of fixed rank n and increasing step. In this case, the limit space is in correspondence with the space of signatures of rectifiable paths in ℝn, as introduced by Chen. Hambly-Lyons’s result on the uniqueness of signature implies that this space is a geodesic metric tree. As a particular consequence we deduce that every path in ℝn can be approximated by projections of some geodesics in some Carnot group of rank n, giving an evidence that the complexity of sub-Riemannian geodesics increases with the step. peerReviewed

Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed to be Borel, or even analytic. One of the theorems states that if $K \subset \mathbb{R}^{n}$ is any set with equal Hausdorff and packing dimensions, then $$ \dim_{\mathrm{H}} π_{e}(K) = \min\{\dim_{\mathrm{H}} K,1\} $$ for almost every $e \in S^{n - 1}$. Here $π_{e}$ stands for orthogonal projection to $\mathrm{span}(e)$. The primary purpose of this paper is to present proofs for Lutz and Stull's projection theorems which do not refer to information theoretic concepts. Instead, they will rely on combinatorial-geometric arguments, such as discretised versions of Kaufman's "potential theoretic" method, the pigeonhole principle, and a lemma of Katz and Tao. A secondary purpose is to slightly generalise Lutz and Stull's theorems: the versions in this paper apply to orthogonal projections to $m$-planes in $\mathbb{R}^{n}$, for all $0 < m < n$., 11 pages. v2: Incorporated referee suggestions

We study a measure-theoretic notion of connectedness for sets of finite perimeter in the setting of doubling metric measure spaces supporting a weak $(1,1)$-Poincar\'{e} inequality. The two main results we obtain are a decomposition theorem into indecomposable sets and a characterisation of extreme points in the space of BV functions. In both cases, the proof we propose requires an additional assumption on the space, which is called isotropicity and concerns the Hausdorff-type representation of the perimeter measure., Comment: 32 pages

This paper is related to the problem of finding a good notion of rectifiability in sub-Riemannian geometry. In particular, we study which kind of results can be expected for smooth hypersurfaces in Carnot groups. Our main contribution will be a consequence of the following result: there exists a -hypersurface without characteristic points that has uncountably many pairwise non-isomorphic tangent groups on every positive-measure subset. The example is found in a Carnot group of topological dimension 8, it has Hausdorff dimension 12 and so we use on it the Hausdorff measure . As a consequence, we show that any Lipschitz map defined on a subset of a Carnot group of Hausdorff dimension 12, with values in , has negligible image with respect to the Hausdorff measure . In particular, we deduce that cannot be Lipschitz parametrizable by countably many maps each defined on some subset of some Carnot group of Hausdorff dimension 12. As main consequence we have that a notion of rectifiability proposed by S. Pauls is not equivalent to one proposed by B. Franchi, R. Serapioni and F. Serra Cassano, at least for arbitrary Carnot groups. In addition, we show that, given a subset of a homogeneous subgroup of Hausdorff dimension 12 of a Carnot group, every bi-Lipschitz map satisfies . Finally, we prove that such an example does not exist in Heisenberg groups: we prove that all -hypersurfaces in with are countably -rectifiable according to Pauls’ definition, even with bi-Lipschitz maps. peerReviewed

The purpose of this note is to record a consequence, for general metric spaces, of a recent result of David Bate. We prove the following fact: Let $X$ be a compact metric space of topological dimension $n$. Suppose that the $n$-dimensional Hausdorff measure of $X$, $\mathcal H^n(X)$, is finite. Suppose further that the lower n-density of the measure $\mathcal H^n$ is positive, $\mathcal H^n$-almost everywhere in $X$. Then $X$ contains an $n$-rectifiable subset of positive $\mathcal H^n$-measure. Moreover, the assumption on the lower density is unnecessary if one uses recently announced results of Cs\"ornyei-Jones., Comment: 5 pages, Comments welcome

We consider submanifolds of sub-Riemannian Carnot groups with intrinsic $$C^1$$ C 1 regularity ($$C^1_H$$ C H 1 ). Our first main result is an area formula for $$C^1_H$$ C H 1 intrinsic graphs; as an application, we deduce density properties for Hausdorff measures on rectifiable sets. Our second main result is a coarea formula for slicing $$C^1_H$$ C H 1 submanifolds into level sets of a $$C^1_H$$ C H 1 function.

Suppose that E⊂Rn+1 is a uniformly rectifiable set of codimension 1. We show that every harmonic function is ε-approximable in Lp(Ω) for every p∈(1,∞), where Ω:=Rn+1∖E. Together with results of many authors this shows that pointwise, L∞ and Lp type ε-approximability properties of harmonic functions are all equivalent and they characterize uniform rectifiability for codimension 1 Ahlfors–David regular sets. Our results and techniques are generalizations of recent works of T. Hytönen and A. Rosén and the first author, J. M. Martell and S. Mayboroda. peerReviewed

In this paper, we prove that a metric measure space which has at least one open set isometric to an interval, and for which the (possibly non-unique) optimal transport map exists from any absolutely continuous measure to an arbitrary measure, is a one-dimensional manifold (possibly with boundary). As an immediate corollary we obtain that if a metric measure space is a very strict $CD(K,N)$ -space or an essentially non-branching $MCP(K,N)$-space with some open set isometric to an interval, then it is a one-dimensional manifold. We also obtain the same conclusion for a metric measure space which has a point in which the Gromov-Hausdorff tangent is unique and isometric to the real line, and for which the optimal transport maps not only exist but are unique. Again, we obtain an analogous corollary in the setting of essentially non-branching $MCP(K,N)$-spaces., 15 pages, 2 figures, to appear in Proceedings of the AMS

Applying the Bonk-Kleiner characterization of Ahlfors 2-regular quasispheres, we show that a metric two-sphere $X$ is a quasisphere if and only if $X$ is linearly locally connected and carries a weak metric doubling measure, i.e., a measure that deforms the metric on $X$ without much shrinking., Comment: 15 pages

We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces $X$ homeomorphic to $\mathbb R^2$. Given a measure $\mu$ on such a space, we introduce $\mu$-quasiconformal maps $f:X \to \mathbb R^2$, whose definition involves deforming lengths of curves by $\mu$. We show that if $\mu$ is an infinitesimally metric measure, i.e., it satisfies an infinitesimal version of the metric doubling measure condition of David and Semmes, then such a $\mu$-quasiconformal map exists. We apply this result to give a characterization of the metric spaces admitting an infinitesimally quasisymmetric parametrization., Comment: 25 pages, 1 figure

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A metric measure space $(X,d,\mu)$ is said to satisfy the strong annular decay condition if there is a constant $C>0$ such that $$ \mu\big(B(x,R)\setminus B(x,r)\big)\leq C\, \frac{R-r}{R}\, \mu (B(x,R)) $$ for each $x\in X$ and all $0, Comment: 13 pages

A family of planar curves is called a Moser family if it contains an isometric copy of every rectifiable curve in $\mathbb{R}^{2}$ of length one. The classical "worm problem" of L. Moser from 1966 asks for the least area covered by the curves in any Moser family. In 1979, J. M. Marstrand proved that the answer is not zero: the union of curves in a Moser family has always area at least $c$ for some small absolute constant $c > 0$. We strengthen Marstrand's result by showing that for $p > 3$, the $p$-modulus of a Moser family of curves is at least $c_{p} > 0$., 18 pages, 3 figures. v2: Improved main result (replaced 4 by 3)

We introduce a more restrictive version of the strict CD(K,∞) -condition, the so-called very strict CD(K,∞) -condition, and show the existence of optimal maps in very strict CD(K,∞) -spaces despite the possible lack of uniqueness of optimal plans. peerReviewed

Only abstract. Paper copies of master’s theses are listed in the Helka database (http://www.helsinki.fi/helka). Electronic copies of master’s theses are either available as open access or only on thesis terminals in the Helsinki University Library. Vain tiivistelmä. Sidottujen gradujen saatavuuden voit tarkistaa Helka-tietokannasta (http://www.helsinki.fi/helka). Digitaaliset gradut voivat olla luettavissa avoimesti verkossa tai rajoitetusti kirjaston opinnäytekioskeilla. Endast sammandrag. Inbundna avhandlingar kan sökas i Helka-databasen (http://www.helsinki.fi/helka). Elektroniska kopior av avhandlingar finns antingen öppet på nätet eller endast tillgängliga i bibliotekets avhandlingsterminaler. Tämän opinnäytetyön tarkoituksena on perehtyä viime vuosina kehitettyihin tapoihin approksimoida suuria, anonyymeja, normaalimuotoisia pelejä. Peliteoriassa peliä kutsutaan suureksi, jos sen pelaajajoukko on suuri, ja anonyymiksi, jos jokaisen pelaajan hyöty riippuu vain hänen omasta valinnastaan sekä jokaisen sallitun strategian valinneiden vastustajien lukumääristä. Toisin sanoen pelaajan näkökulmasta anonyymin pelin vastustajat eivät poikkea merkittävästi toisistaan ja yksittäisen vastustajan identiteetillä ei ole siis vaikutusta pelaajan pelistä kokemaan hyötyyn. Pelaajamäärän kasvaessa pelin mahdollisten lopputulosten joukko kasvaa eksponentiaalisesti suhteessa pelaajajoukkoon. Yleensä tämä vaikeuttaa huomattavasti pelin Nash-tasapainojen ratkaisemista. Suuriin anonyymeihin peleihin on kuitenkin kehitetty niiden approksimoimiseen tarkoitettuja, helpommin ratkeavia malleja. Tämä tutkielma keskittyy erityisesti tarkastelemaan Nabil Al-Najjarin vuonna 2008 konstruoimia diskreettejä suuria pelejä. Diskreeteissä suurissa peleissä pelaajajoukko oletetaan numeroituvasti äärettömäksi ja pelaajajoukkojen kokoja mitataan äärellisesti additiivisella mitalla. Näin päästään operoimaan matemaattisen mitta- ja integrointiteorian välineillä ja mallien ratkaiseminen helpottuu. Toisaalta äärellisesti additiivinen mitta ratkaisee aiempia malleja, erityisesti David Schmeidlerin vuonna 1973 esittämää kontinuumipeliä vaivanneet tekniset mitallisuusongelmat. Keskeiseen osaan tutkielmassa nousee suurten lukujen lain versio, joka vähentää pelaajien diskreetissä suuressa pelissä kokemaa epävarmuutta pelin tuloksesta. Työssä todistetaan sen avulla mm. tuloksia, jotka takaavat, että tavallisimmissa tilanteissa diskreetit suuret pelit approksimoivat hyvin vastaavia anonyymeja, normaalimuotoisia pelejä, joissa pelaajajoukko on erittäin suuri mutta äärellinen. Työtä kirjoittaessani huomasin valitettavasti Al-Najjarin alkuperäistekstistä joitain virheitä, joiden johdosta eräitä tuloksia oli heikennettävä (ks. lauseet 5 ja 9). Annan lisäksi joillekin alkuperäistekstin virheellisistä todistuksista uuden korjatun todistuksen. Työssä esitettyjen lauseiden todistukset muodostavat keskeisen sisällön kirjoittajan matematiikan pro gradu -tutkielmasta, mutta ne on täydellisyyden vuoksi sisällytetty myös tämän työn liitteisiin.