|
|
| Line 1: |
Line 1: |
| | + | {{MSC|03E04}} ''in set theory'' |
| | | | |
| − | {{MSC|26A45}} (Functions of one variable) | + | {{MSC|28A}} ''in measure theory'' |
| | | | |
| − | {{MSC|26B30|28A15,26B15,49Q15}} (Functions of severable variables)
| + | ==Set theory== |
| − | | + | A minimal non-zero element of a [[Partially ordered set|partially ordered set]] with a zero $0$, i.e. an element $p$ such that $0<x\leq p$ implies $x=p$. |
| − | [[Category:Analysis]] | + | ==Measure algebras== |
| − | | + | For the definition and relevance in the theory of measure algebras we refer to [[Measure algebra]]. |
| − | {{TEX|done}}
| + | ==Classical measure theory== |
| − | | + | ===Definition=== |
| − | Also called ''total variation''. A numerical characteristic of functions of one or more real variables which is connected with differentiability properties.
| + | Let $\mu$ be a (nonnegative) [[Measure|measure]] on a [[Algebra of sets|$\sigma$-algebra]] $\mathcal{S}$ of subsets of a set $X$. An element $a\in \mathcal{S}$ is called an ''atom'' of $\mu$ if |
| − | | + | *$\mu (A)>0$; |
| − | ==Functions of one variable==
| + | *For every $B\in \mathcal{S}$ with $B\subset A$ either $\mu (B)=0$ or $\mu (B)=\mu (A)$ |
| − | ===Classical definition===
| + | (cp. with Section IV.9.8 of {{Cite|DS}} or {{Cite|Fe}}). |
| − | Let $I\subset \mathbb R$ be an interval. The total variation is defined in the following way.
| |
| − | | |
| − | '''Definition 1'''
| |
| − | Consider the collection $\Pi$ of ordered $(N+1)$-ples of points $a_1<a_2 < \ldots < a_{N+1}\in I$,
| |
| − | where $N$ is an arbitrary natural number. The total variation of a function $f: I\to \mathbb R$ is given by
| |
| − | \begin{equation}\label{e:TV}
| |
| − | TV\, (f) := \sup \left\{ \sum_{i=1}^N |f(a_{i+1})-f(a_i)| : (a_1, \ldots, a_{N+1})\in\Pi\right\}\,
| |
| − | \end{equation}
| |
| − | (cp. with Section 4.4 of {{cite|Co}} or Section 10.2 of {{Cite|Ro}}).
| |
| − | | |
| − | If the total variation is finite, then $f$ is called a [[Function of bounded variation|function of bounded variation]]. For examples, properties and issues related to the space of functions of bounded variation we refer to [[Function of bounded variation]].
| |
| − | | |
| − | The definition of total variation of a function of one real variable can be easily generalized when the target is a [[Metric space|metric space]] $(X,d)$: it suffices to substitute $|f(a_{i+1})-f(a_i)|$ with $d (f(a_{i+1}), f(a_i))$ in \ref{e:TV}.
| |
| − | | |
| − | ===Modern definition and relation to measure theory=== | |
| − | Classically right-continuous functions of bounded variations can be mapped one-to-one to [[Signed measure|signed measures]]. More precisely, consider a signed measure $\mu$ on (the [[Borel set|Borel subsets ]] of) $\mathbb R$ with finite total variation (see [[Signed measure]] for the definition). We then define the function
| |
| − | \begin{equation}\label{e:F_mu}
| |
| − | F_\mu (x) := \mu (]-\infty, x])\, .
| |
| − | \end{equation} | |
| − | | |
| − | '''Theorem 2'''
| |
| − | * For every signed measure $\mu$ with finite total variation, $F_\mu$ is a right-continuous function of bounded variation such that $\lim_{x\to -\infty} F_\mu (x) = 0$ and $TV (f)$ equals the total variation of $\mu$ (i.e. $|\mu| (\mathbb R))$.
| |
| − | * For every right-continuous function $f:\mathbb R\to \mathbb R$ of bounded variation with $\lim_{x\to-\infty} f (x) = 0$ there is a unique signed measure $\mu$ such that $f=F_\mu$. | |
| − | Moreover, the total variation of $f$ equals the total variation of the measure $\mu$ (cp. with [[Signed measure]] for the definition).
| |
| − | | |
| − | For a proof see Section 4 of Chapter 4 in {{Cite|Co}}. Obvious generalizations hold in the case of different domains of definition.
| |
| | | | |
| | + | '''Remark''' |
| | + | If we denote by $\mathcal{N}$ the null sets and consider the standard quotient measure algebra $(\mathcal{S}/\mathcal{N}, \mu)$, then any atom of such quotient measure algebra corresponds to an equivalence class of atoms of $\mu$. |
| | + | ===Atomic measures=== |
| | + | A measure $\mu$ is called ''atomic'' if there is a partition of $X$ into countably many elements of $\mathcal{A}$ which are either atoms or null sets. An atomic probability neasure is often called ''atomic distribution''. Examples of atomic distributions are the [[Discrete distribution|discrete distributions]]. |
| | + | ===Nonatomic measures=== |
| | + | A measure $\mu$ is called ''nonatomic'' it has no atoms. |
| | ===Jordan decomposition=== | | ===Jordan decomposition=== |
| − | {{Anchor|Jordan decomposition}}
| + | If $\mu$ is $\sigma$-finite, it is possible to decompose $\mu$ as $\mu_a+\mu_{na}$, where $\mu_a$ is an atomic measure and $\mu_{na}$ is a nonatomic measure. In case $\mu$ is a probability measure, this means that $\mu$ can be written as $p \mu_a + (1-p) \mu_{na}$, where $p\in [0,1]$, $\mu_a$ is an atomic probability measure and $\mu_{na}$ a nonatomic probability measure (see {{Cite|Fe}}), which is sometimes called a [[Continuous distribution|continuous distribution]]. This decomposition is sometimes called ''Jordan decomposition'', although several authors use this name in other contexts, see [[Jordan decomposition]]. |
| − | A fundamental characterization of functions of bounded variation of one variable is due to Jordan.
| + | ===Measures in the euclidean space=== |
| − | | + | If $\mu$ is a $\sigma$-finite measure on the [[Borel set|Borel $\sigma$-algebra]] of $\mathbb R^n$, then it is easy to show that, for any atom $B$ of $\mu$ there is a point $x\in B$ with the property that $\mu (B) = \mu (\{x\})$. Thus such a measure is atomic if and only if it is the countable sum of [[Delta-function|Dirac deltas]], i.e. if there is an (at most) countable set $\{x_i\}\subset \mathbb R^n$ and an (at most) countable set $\{\alpha_i\}\subset ]0, \infty[$ with the property that |
| − | '''Theorem 3'''
| |
| − | Let $I\subset \mathbb R$ be an interval. A function $f: I\to\mathbb R$ has bounded variation if and only if it can be written as the difference of two bounded nondecreasing functions.
| |
| − | | |
| − | (Cp. with Theorem 4 of Section 5.2 in {{Cite|Ro}}). Indeed it is possible to find a canonical representation of any function of bounded variation as difference of nondecreasing functions.
| |
| − | | |
| − | '''Theorem 4'''
| |
| − | If $f:[a,b] \to\mathbb R$ is a function of bounded variation then there is a pair of nondecreasing functions $f^+$ and $f^-$ such that $f= f^+- f^-$ and $TV (f) = f^+ (b)-f^+ (a) + f^- (b)- f^- (a)$. The pair is unique up to addition of a constant, i.e. if $g^+$ and $g^-$ is a second pair with the same property, then $g^+-g^-=f^+-f^-\equiv {\rm const}$.
| |
| − | | |
| − | (Cp. with Theorem 3 of Section 5.2 in {{Cite|Ro}}). The latter representation of a function of bounded variation is also called [[Jordan decomposition]].
| |
| − | ====Negative and positive variations====
| |
| − | It is possible to define the negative and positive variations of $f$ in the following way.
| |
| − | | |
| − | '''Definition 5'''
| |
| − | Let $I\subset \mathbb R$ be an interval and $\Pi$ be as in '''Definition 1'''. The negative and positive variations of $f:I\to\mathbb R$ are then defined as
| |
| − | \[
| |
| − | TV^+ (f):= \sup \left\{ \sum_{i=1}^N \max \{(f(a_{i+1})-f(a_i)), 0\} : (a_1, \ldots, a_{N+1})\in\Pi\right\}\,
| |
| − | \]
| |
| − | \[
| |
| − | TV^- (f) := \sup \left\{ \sum_{i=1}^N \max \{-(f(a_{i+1})-f(a_i)), 0\} : (a_1, \ldots, a_{N+1})\in\Pi\right\}\, .
| |
| − | \]
| |
| − | | |
| − | If $f$ is a function of bounded variation on $[a,b]$ we can define $f^+ (x) = TV^+ (f|_{[a,x]})$ and $f^- (x) = TV^- (f|_{[a,x]})$. Then it turns out that, up to constants, these two functions give the Jordan decomposition of '''Theorem 4''', cp. with Lemma 3 in Section 2, Chapter 5 of {{Cite|Ro}}.
| |
| − | ===Historical remark===
| |
| − | The variation of a function of one real variable was considered for the first time by C. Jordan in {{Cite|Jo}} to study the pointwise convergence of [[Fourier series]], cp. with [[Jordan criterion]] and [[Function of bounded variation]].
| |
| − | ==Wiener's and Young's generalizations==
| |
| − | One sometimes also considers classes $BV_\Phi ([a,b])$ defined as follows. Let $\Phi: [0, \infty[\to [0, \infty[$ be a continuous function with $\Phi (0)=0$ which increases monotonically. If $f:[a,b]\to \mathbb R$, we let $TV_{\phi} (f)$ be the least upper bound of sums of the type
| |
| − | \[
| |
| − | \sum_{i=1}^N \Phi (|f (x_{i+1}- f(x_i)|)
| |
| − | \]
| |
| − | where $a\leq x_1 < \ldots < x_{N+1}<b$ is an arbitrary family of points. The quantity $TV_\Phi (f)$ is called the $\Phi$-variation of $f$ on $[a,b]$. If $TV_\Phi (f)<\infty$ one says that $f$ has bounded $\Phi$-variation on $[a,b]$, while the class of such functions is denoted by $BV_\Phi ([a,b])$ (see {{Cite|Bar}}). If $\Phi (u)=u$, one obtains Jordan's class $BV ([a,b])$, while if $\Phi (u)=u^p$, one obtains Wiener's classes $BV_p ([a,b])$ (see {{Cite|Wi}}). The definition of the class $BV_\Phi ([a,b])$ was proposed by L.C. Young in {{Cite|Yo}}.
| |
| − | | |
| − | If
| |
| − | \[
| |
| − | \limsup_{u\to 0^+} \frac{\Phi_1 (u)}{\Phi_2 (u)} < \infty
| |
| − | \]
| |
| − | then
| |
| − | \[
| |
| − | BV_{\Phi_2} ([a,b])\subset BV_{\Phi_1} ([a,b)]\, .
| |
| − | \]
| |
| − | In particular, on any interval $[a,b]$,
| |
| − | \[
| |
| − | BV_p ([a,b])\subset BV_q ([a,b]) \subset BV_{\exp (-u^{-\alpha})} ([a,b]) \subset
| |
| − | BV_{\exp (-u^{-\beta})} ([a,b])\, .
| |
| − | \]
| |
| − | for $1\leq p < q$ and $0<\alpha<\beta<\infty$, these being proper inclusions.
| |
| − | | |
| − | ==Functions of several variables==
| |
| − | ===Historical remarks===
| |
| − | After the introduction by Jordan of functions of bounded variations of one real variable, several authors attempted to generalize the concept to functions of more than one variable. The first attempt was made by Arzelà and Hardy in 1905, see [[Arzelà variation]] and [[Hardy variation]], followed by Vitali, Fréchet, Tonelli and Pierpont, cp. with [[Vitali variation]], [[Fréchet variation]], [[Tonelli plane variation]] and [[Pierpont variation]] (moreover, the definition of Vitali variation was also considered independently by Lebesgue and De la Vallée-Poussin). However, the point of view which became popular and it is nowadays accepted in the literature as most efficient generalization of the $1$-dimensional theory is due to De Giorgi and Fichera (see {{Cite|DG}} and {{Cite|Fi}}). Though with different definitions, the approaches by De Giorgi and Fichera are equivalent (and very close in spirit) to the ''distributional theory'' described below. A promiment role in the further developing of the theory was also played by Fleming, Federer and Volpert. Moreover, Krickeberg and Fleming showed, independently, that the current definition of functions of bounded variation is indeed equivalent to a slight modification of Tonelli's one {{Cite|To}}, proposed by Cesari {{Cite|Ce}}, cp. with the section '''Tonelli-Cesari variation''' of [[Function of bounded variation]]. We refer to Section 3.12 of {{Cite|AFP}} for a thorough discussion of the topic.
| |
| − | ====Link to the theory of currents==== | |
| − | Functions of bounded variation in $\mathbb R^n$ can be identified with $n$-dimensional normal [[Current|currents]] in $\mathbb R^n$. This is the point of view of Federer, {{Cite|Fe}}, which thus derives most of the conclusions of the theory of $BV$ functions as special cases of more general theorems for normal currents,
| |
| − | | |
| − | ===Definition===
| |
| − | Following {{Cite|EG}}:
| |
| − | | |
| − | '''Definition 6'''
| |
| − | Let $\Omega\subset \mathbb R^n$ be open. The total variation of $u\in L^1_{loc} (\Omega)$ is given by
| |
| − | \begin{equation}\label{e:diverg}
| |
| − | V (u, \Omega) := \sup \left\{ \int_\Omega u\, {\rm div}\, \psi : \psi\in C^\infty_c (\Omega, \mathbb R^n), \, \|\psi\|_{C^0}\leq 1\right\}\, .
| |
| − | \end{equation} | |
| − | If $V(u, \Omega)< \infty$ then we say that $u$ has '''bounded variation'''. The space of functions $u\in L^1 (\Omega)$ which have bounded variation are denoted by $BV (\Omega)$.
| |
| − | | |
| − | As a consequence of the [[Radon-Nikodym theorem]] we then have
| |
| − | | |
| − | '''Prposition 7'''
| |
| − | A function $u\in L^1_{loc} (\Omega)$, then $V(u, \Omega)<\infty$ if and only the [[Generalized derivative|distributional derivative]] of $f$ is a Radon measure $Du$. Moreover $V (u,\Omega) = |Du| (\Omega)$, where $|Du|$ is the total variation measure of $Du$.
| |
| − | | |
| − | When $n=1$ and $\Omega=[a,b]$, then $V (u, [a,b])<\infty$ if and only if there exists a function $\tilde{u}$ such that
| |
| − | $u=\tilde{u}$ a.e. and $TV (\tilde{u})<\infty$. Moreover,
| |
| | \[ | | \[ |
| − | V (u, [a,b]) = \inf \{TV (\tilde{u}): \tilde{u}= t \quad\mbox{a.e.}\}\, .
| + | \mu (A) = \sum_{x_i\in A} \alpha_i \qquad \mbox{for every Borel set $A$}. |
| | \] | | \] |
| − | (Cp. with [[Function of bounded variation]]).
| + | ===Sierpinski's theorem=== |
| − | | + | A nonatomic measure takes a continuum of values. This is a corollary of the following Theorem due to Sierpinski (see {{Cite|Si}}): |
| − | ===Caccioppoli sets=== | |
| − | A special class of $BV_{loc}$ functions which play a fundamental role in the theory (and had also a pivotal role in its historical development) is the set of those $f\in BV$ which takes only the values $0$ and $1$ and are, therefore, the indicator functions of a set. | |
| − | | |
| − | '''Definition 8'''
| |
| − | Let $\Omega\subset \mathbb R^n$ be an open set and $E\subset \Omega$ a measurable set such that $ V({\bf 1}_E, \Omega)<\infty$. The $E$ is called a ''Caccioppoli set'' or a ''set of finite perimeter'' and its perimeter in $\Omega$ is defined to be
| |
| − | \[
| |
| − | {\rm Per}\, (E, \Omega) = V ({\bf 1}_E, \Omega)\, .
| |
| − | \]
| |
| − | | |
| − | | |
| − | ==Coarea formula==
| |
| − | An important tool which allows often to reduce problems for $BV$ functions to problems for Caccioppoli sets is the following generalization of the [[Coarea formula]], first proved by Fleming and Rishel in {{Cite|FR}}.
| |
| − | | |
| − | '''Theorem 29'''
| |
| − | For any open set $\Omega\subset \mathbb R^n$ and any $u\in L^1 (\Omega)$, the map $t\mapsto {\rm Per}\, (\{u>t\}, \Omega)$ is Lebesgue measurable and one has
| |
| − | \[
| |
| − | V (u,\Omega) = \int_{-\infty}^\infty {\rm Per}\, (\{u>t\}, \Omega)\, dt\,
| |
| − | \]
| |
| − | In particular, if $u\in BV (\Omega)$, then $U_t:=\{u>t\}$ is a Caccioppoli set for a.e. $t$ and, for any Borel set $B\subset \Omega$,
| |
| − | \[
| |
| − | |Du| (B) = \int_{-\infty}^\infty |D{\bf 1}_{U_t}| (B)\, dt \qquad\mbox{and}\qquad
| |
| − | Du (B) = \int_{-\infty}^\infty D{\bf 1}_{U_t} (B)\, dt\,
| |
| − | \]
| |
| − | (where the maps $t\mapsto |D{\bf 1}_{U_t}| (B)$ and $t\mapsto D{\bf 1}_{U_t} (B)$ are both Lebesgue measurable).
| |
| − | | |
| − | Cp. with Theorem 3.40 in {{Cite|AFP}}.
| |
| − | ===Banach indicatrix===
| |
| − | Let $f:[a,b]\to \mathbb R$. The [[Banach indicatrix]] $N (y,f)$ is then the cardinality of the set $\{f=y\}$. A special
| |
| − | case of the coarea formula, first proved by Banach in {{Cite|Ba}}, is the identity
| |
| − | \[
| |
| − | TV (f) = \int_{-\infty}^\infty N (f,y)\, dy
| |
| − | \]
| |
| − | which holds for every '''continuous''' function $f:[a,b]\to\mathbb R$.
| |
| − | | |
| − | ===Vitushkin variation===
| |
| − | In {{Cite|Vi}} Vitushkin proposed a notion of variation for functions of several variables based on the [[Banach indicatrix|Banach indicatrix]]. Let $f: [0,1]^n\to \mathbb R$ be a Lebesgue measurable function and $k\in \{1, \ldots, n\}$. The variation $V_k (f)$ of order $k$ of $f$ on $[0,1]^n$ is the number
| |
| − | \[
| |
| − | \int_{-\infty}^\infty v_{k-1} (\{f=t\})\, dt
| |
| − | \]
| |
| − | where $v_{k-1} (E)$ denotes the [[Variation of a set|Variation of the set]] $E$ of order $k-1$. The Vitushkin variation
| |
| − | enjoys the following properties:
| |
| − | | |
| − | a) $V_n (f+g)\leq V_n (f) + V_n (g)$.
| |
| − | | |
| − | b) If a sequence of functions $f_j$ converges uniformly to $f$ in $[0,1]^n$, then
| |
| − | \[
| |
| − | V_k (f) \leq \liminf_{j\to\infty}\; V_k (f_j)\, .
| |
| − | \]
| |
| − | | |
| − | c) If the function $f$ is continuous and all its variations are finite, then $f$ has a total differential almost-everywhere.
| |
| − | | |
| − | d) If the function $f$ is absolutely continuous then
| |
| − | \[
| |
| − | V_n (f) = \int_{[0,1]^n} |\nabla f (x)|\, dx
| |
| − | \]
| |
| − | and hence coincides with the variation $V (f, [0,1]^n)$. However, the two quantities differ in general.
| |
| − | | |
| − | e) If the function $f$ is continuous, it has bounded variations of all orders and can be extended periodically with period $1$ in all variables, then its Fourier series converges uniformly to it (Pringsheim's theorem).
| |
| − | | |
| − | If the function $f$ has continuous derivatives of all orders up to and including $n-k+1$, then its variation of order $k$ is finite. The smoothness conditions cannot be improved for any $k$.
| |
| − | | |
| − | | |
| − | | |
| | | | |
| | + | '''Theorem''' |
| | + | If $\mu$ is a nonatomic measure on a $\sigma$-algebra $\mathcal{A}$ and $A\in \mathcal{A}$ an element such that $\mu (A)>0$, then for every $b\in [0, \mu (B)]$ there is an element $B\in \mathcal{A}$ with $B\subset A$ and $\mu (B) = b$. |
| | | | |
| | + | ==Comment== |
| | + | By a natural extension of meaning, the term atom is also used for an object of a category having no subobjects other than itself and the null subobject (cf. [[Null object of a category|Null object of a category]]). |
| | | | |
| | ==References== | | ==References== |
| | {| | | {| |
| | |- | | |- |
| − | |valign="top"|{{Ref|AFP}}|| L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. {{MR|1857292}}{{ZBL|0957.49001}} | + | |valign="top"|{{Ref|DS}}|| N. Dunford, J.T. Schwartz, "Linear operators. General theory", '''1''', Interscience (1958). {{MR|0117523}} {{ZBL|0635.47001}} |
| − | |-
| |
| − | |valign="top"|{{Ref|Ba}}|| S. Banach, "Sur les lignes rectifiables et les surfaces dont l'aire est finie" ''Fund. Math.'' , '''7''' (1925) pp. 225–236.
| |
| − | |-
| |
| − | |valign="top"|{{Ref|Bar}}|| N.K. Bary, "A treatise on trigonometric series" , Pergamon (1964).
| |
| − | |-
| |
| − | |valign="top"|{{Ref|Ca}}|| R. Caccioppoli, "Misura e integrazione sugli insiemei dimensionalmente orientati I, II", Rend. Acc. Naz. Lincei (8), {\bf 12} (1952) pp. 3-11 and 137-146.
| |
| − | |-
| |
| − | |valign="top"|{{Ref|Co}}|| D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
| |
| − | |-
| |
| − | |valign="top"|{{Ref|DG}}|| E. De Giorgi, "Su una teoria generale della misura $n-1$-dimensionale in uno spazio a $r$ dimensioni", Ann. Mat. Pura Appl. (4), '''36''' (1954) pp. 191-213.
| |
| − | |-
| |
| − | |valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. {{MR|1158660}} {{ZBL|0804.2800}}
| |
| − | |-
| |
| − | |valign="top"|{{Ref|Fe}}|| H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969. {{MR|0257325}} {{ZBL|0874.49001}}
| |
| − | |-
| |
| − | |valign="top"|{{Ref|Fi}}|| G. Fichera, "Lezioni sulle trasformazioni lineari", Istituto matematico dell'Università di Trieste, vol. I, 1954.
| |
| − | |-
| |
| − | |valign="top"|{{Ref|FR}}|| W. H. Fleming, R. Rishel, "An integral formula for total gradient variation", Arch. Math., '''11''' (1960) pp. 218-222.
| |
| − | |-
| |
| − | |valign="top"|{{Ref|Jo}}|| C. Jordan, "Sur la série de Fourier" ''C.R. Acad. Sci. Paris'' , '''92''' (1881) pp. 228–230
| |
| − | |-
| |
| − | |valign="top"|{{Ref|Ro}}|| H.L. Royden, "Real analysis" , Macmillan (1969). {{MR|0151555}} {{ZBL|0197.03501}}
| |
| − | |-
| |
| − | |valign="top"|{{Ref|To}}|| L. Tonelli, "Sulle funzioni di due variabili generalmente a variazione limitata", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2), '''5''' (1936) pp. 315-320.
| |
| | |- | | |- |
| − | |valign="top"|{{Ref|Vi}}|| A.G. Vitushkin, "On higher-dimensional variations", Moscow (1955). | + | |valign="top"|{{Ref|Fe}}|| W. Feller, "An introduction to probability theory and its applications"|"An introduction to probability theory and its applications", '''2''', Wiley (1971). |
| | |- | | |- |
| − | |valign="top"|{{Ref|Wi}}|| N. Wiener, "The quadratic variation of a function and its Fourier coefficients" ''J. Math. and Phys.'' , '''3''' (1924) pp. 72–94. | + | |valign="top"|{{Ref|Lo}}|| M. Loève, "Probability theory", Princeton Univ. Press (1963). {{MR|0203748}} {{ZBL|0108.14202}} |
| | |- | | |- |
| − | |valign="top"|{{Ref|Yo}}|| L.C. Young, "Sur une généralisation de la notion de variation de puissance $p$ borneé au sens de M. Wiener, et sur la convergence des series de Fourier" ''C.R. Acad. Sci. Paris Sér. I Math.'' , '''204''' (1937) pp. 470–472 | + | |valign="top"|{{Ref|Si}}|| W. Sierpinski, "Sur le fonctions d'enseble additives et continuoes", '''3''', Fund. Math. (1922) pp. 240-246. |
| | |- | | |- |
| | |} | | |} |