Misplaced Pages

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67-585: Henri Lebesgue described this measure in the year 1901 which, a year after, was followed up by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902. For any interval I = [ a , b ] {\displaystyle I=[a,b]} , or I = ( a , b ) {\displaystyle I=(a,b)} , in the set R {\displaystyle \mathbb {R} } of real numbers, let ℓ ( I ) = b −

134-494: A {\displaystyle \ell (I)=b-a} denote its length. For any subset E ⊆ R {\displaystyle E\subseteq \mathbb {R} } , the Lebesgue outer measure λ ∗ ( E ) {\displaystyle \lambda ^{\!*\!}(E)} is defined as an infimum The above definition can be generalised to higher dimensions as follows. For any rectangular cuboid C {\displaystyle C} which

201-438: A , b ) {\displaystyle (a,b)} is a 1-dimensional open ball with a center at 1 2 ( a + b ) {\displaystyle {\tfrac {1}{2}}(a+b)} and a radius of 1 2 ( b − a ) . {\displaystyle {\tfrac {1}{2}}(b-a).} The closed finite interval [ a , b ] {\displaystyle [a,b]}

268-406: A , b ) {\displaystyle [a,b)} are neither an open set nor a closed set. If one allows an endpoint in the closed side to be an infinity (such as (0,+∞] ), the result will not be an interval, since it is not even a subset of the real numbers. Instead, the result can be seen as an interval in the extended real line , which occurs in measure theory , for example. In summary,

335-442: A real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity , indicating the interval extends without a bound . A real interval can contain neither endpoint, either endpoint, or both endpoints, excluding any endpoint which is infinite. For example, the set of real numbers consisting of 0 , 1 , and all numbers in between

402-664: A σ -algebra . A set E {\displaystyle E} that does not satisfy the Carathéodory criterion is not Lebesgue-measurable. ZFC proves that non-measurable sets do exist; an example is the Vitali sets . The first part of the definition states that the subset E {\displaystyle E} of the real numbers is reduced to its outer measure by coverage by sets of open intervals. Each of these sets of intervals I {\displaystyle I} covers E {\displaystyle E} in

469-450: A and b are real numbers such that a ≤ b : {\displaystyle a\leq b\colon } The closed intervals are those intervals that are closed sets for the usual topology on the real numbers. The empty set and R {\displaystyle \mathbb {R} } are the only intervals that are both open and closed. A half-open interval has two endpoints and includes only one of them. It

536-448: A and b included. The notation [ a .. b ] is used in some programming languages ; in Pascal , for example, it is used to formally define a subrange type, most frequently used to specify lower and upper bounds of valid indices of an array . Another way to interpret integer intervals are as sets defined by enumeration , using ellipsis notation. An integer interval that has

603-493: A half-space is taken as a kind of degenerate ball (without a well-defined center or radius), a half-space can be taken as analogous to a half-bounded interval, with its boundary plane as the (degenerate) sphere corresponding to the finite endpoint. A finite interval is (the interior of) a 1-dimensional hyperrectangle . Generalized to real coordinate space R n , {\displaystyle \mathbb {R} ^{n},} an axis-aligned hyperrectangle (or box)

670-423: A "nicer" set B which differs from A only by a null set (in the sense that the symmetric difference ( A − B ) ∪ ( B − A ) is a null set) and then show that B can be generated using countable unions and intersections from open or closed sets. The modern construction of the Lebesgue measure is an application of Carathéodory's extension theorem . It proceeds as follows. Fix n ∈ N . A box in R

737-419: A ,  a ) , [ a ,  a ) , and ( a ,  a ] represents the empty set , whereas [ a ,  a ] denotes the singleton set  { a } . When a > b , all four notations are usually taken to represent the empty set. Both notations may overlap with other uses of parentheses and brackets in mathematics. For instance, the notation ( a , b ) is often used to denote an ordered pair in set theory,

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804-491: A , +∞) are all meaningful and distinct. In particular, (−∞, +∞) denotes the set of all ordinary real numbers, while [−∞, +∞] denotes the extended reals. Even in the context of the ordinary reals, one may use an infinite endpoint to indicate that there is no bound in that direction. For example, (0, +∞) is the set of positive real numbers , also written as R + . {\displaystyle \mathbb {R} _{+}.} The context affects some of

871-425: A characterization of measurability in abstract spaces. Indeed, in the generalization to abstract measures, this theorem is sometimes extended to a definition of measurability. Thus, we have the following definition: If μ ∗ : P ( Ω ) → [ 0 , ∞ ] {\displaystyle \mu ^{*}:{\mathcal {P}}(\Omega )\to [0,\infty ]}

938-454: A finite lower or upper endpoint always includes that endpoint. Therefore, the exclusion of endpoints can be explicitly denoted by writing a .. b  − 1  , a  + 1 .. b  , or a  + 1 .. b  − 1 . Alternate-bracket notations like [ a .. b ) or [ a .. b [ are rarely used for integer intervals. The intervals are precisely the connected subsets of R . {\displaystyle \mathbb {R} .} It follows that

1005-410: A maximum or is right unbounded; it is simply closed if it is both left-closed and right closed. So, the closed intervals coincide with the closed sets in that topology. The interior of an interval I is the largest open interval that is contained in I ; it is also the set of points in I which are not endpoints of I . The closure of I is the smallest closed interval that contains I ; which

1072-468: A sense, since the union of these intervals contains E {\displaystyle E} . The total length of any covering interval set may overestimate the measure of E , {\displaystyle E,} because E {\displaystyle E} is a subset of the union of the intervals, and so the intervals may include points which are not in E {\displaystyle E} . The Lebesgue outer measure emerges as

1139-441: A set of the real numbers is an interval, if and only if it is an open interval, a closed interval, or a half-open interval. A degenerate interval is any set consisting of a single real number (i.e., an interval of the form [ a , a ] ). Some authors include the empty set in this definition. A real interval that is neither empty nor degenerate is said to be proper , and has infinitely many elements. An interval

1206-415: A subset X ⊆ R {\displaystyle X\subseteq \mathbb {R} } is also the convex hull of X . {\displaystyle X.} The closure of an interval is the union of the interval and the set of its finite endpoints, and hence is also an interval. (The latter also follows from the fact that the closure of every connected subset of a topological space

1273-567: Is Lebesgue measurable if and only if λ ∗ ( S ) = λ ∗ ( S ∩ M ) + λ ∗ ( S ∩ M c ) {\displaystyle \lambda ^{*}(S)=\lambda ^{*}(S\cap M)+\lambda ^{*}\left(S\cap M^{c}\right)} for every S ⊆ R n , {\displaystyle S\subseteq \mathbb {R} ^{n},} where M c {\displaystyle M^{c}} denotes

1340-707: Is a Cartesian product C = I 1 × ⋯ × I n {\displaystyle C=I_{1}\times \cdots \times I_{n}} of open intervals, let vol ⁡ ( C ) = ℓ ( I 1 ) × ⋯ × ℓ ( I n ) {\displaystyle \operatorname {vol} (C)=\ell (I_{1})\times \cdots \times \ell (I_{n})} (a real number product) denote its volume. For any subset E ⊆ R n {\displaystyle E\subseteq \mathbb {R^{n}} } , Some sets E {\displaystyle E} satisfy

1407-456: Is a closed set of the real line , but an interval that is a closed set need not be a closed interval. For example, intervals ( − ∞ , b ] {\displaystyle (-\infty ,b]} and [ a , + ∞ ) {\displaystyle [a,+\infty )} are also closed sets in the real line. Intervals ( a , b ] {\displaystyle (a,b]} and [

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1474-448: Is a bounded real interval whose endpoints are j 2 n {\displaystyle {\tfrac {j}{2^{n}}}} and j + 1 2 n , {\displaystyle {\tfrac {j+1}{2^{n}}},} where j {\displaystyle j} and n {\displaystyle n} are integers. Depending on the context, either endpoint may or may not be included in

1541-516: Is a connected subset.) In other words, we have The intersection of any collection of intervals is always an interval. The union of two intervals is an interval if and only if they have a non-empty intersection or an open end-point of one interval is a closed end-point of the other, for example ( a , b ) ∪ [ b , c ] = ( a , c ] . {\displaystyle (a,b)\cup [b,c]=(a,c].} If R {\displaystyle \mathbb {R} }

1608-411: Is a degenerate interval (see below). The open intervals are those intervals that are open sets for the usual topology on the real numbers. A closed interval is an interval that includes all its endpoints and is denoted with square brackets. For example, [0, 1] means greater than or equal to 0 and less than or equal to 1 . Closed intervals have one of the following forms in which

1675-428: Is a generalization of the Lebesgue measure that is useful for measuring the subsets of R of lower dimensions than n , like submanifolds , for example, surfaces or curves in R and fractal sets. The Hausdorff measure is not to be confused with the notion of Hausdorff dimension . It can be shown that there is no infinite-dimensional analogue of Lebesgue measure . Interval (mathematics) In mathematics ,

1742-403: Is a set of the form where b i ≥ a i , and the product symbol here represents a Cartesian product. The volume of this box is defined to be For any subset A of R , we can define its outer measure λ *( A ) by: We then define the set A to be Lebesgue-measurable if for every subset S of R , These Lebesgue-measurable sets form a σ -algebra , and the Lebesgue measure

1809-441: Is also the set I augmented with its finite endpoints. For any set X of real numbers, the interval enclosure or interval span of X is the unique interval that contains X , and does not properly contain any other interval that also contains X . An interval I is a subinterval of interval J if I is a subset of J . An interval I is a proper subinterval of J if I is a proper subset of J . However, there

1876-638: Is an outer measure on a set Ω , {\displaystyle \Omega ,} where P ( Ω ) {\displaystyle {\mathcal {P}}(\Omega )} denotes the power set of Ω , {\displaystyle \Omega ,} then a subset M ⊆ Ω {\displaystyle M\subseteq \Omega } is called μ ∗ {\displaystyle \mu ^{*}} –measurable or Carathéodory-measurable if for every S ⊆ Ω , {\displaystyle S\subseteq \Omega ,}

1943-408: Is an interval, denoted [0, 1] and called the unit interval ; the set of all positive real numbers is an interval, denoted (0, ∞) ; the set of all real numbers is an interval, denoted (−∞, ∞) ; and any single real number a is an interval, denoted [ a , a ] . Intervals are ubiquitous in mathematical analysis . For example, they occur implicitly in the epsilon-delta definition of continuity ;

2010-449: Is conflicting terminology for the terms segment and interval , which have been employed in the literature in two essentially opposite ways, resulting in ambiguity when these terms are used. The Encyclopedia of Mathematics defines interval (without a qualifier) to exclude both endpoints (i.e., open interval) and segment to include both endpoints (i.e., closed interval), while Rudin's Principles of Mathematical Analysis calls sets of

2077-452: Is considered in the special section below . An interval is a subset of the real numbers that contains all real numbers lying between any two numbers of the subset. The endpoints of an interval are its supremum , and its infimum , if they exist as real numbers. If the infimum does not exist, one says often that the corresponding endpoint is − ∞ . {\displaystyle -\infty .} Similarly, if

Lebesgue measure - Misplaced Pages Continue

2144-410: Is defined by λ ( A ) = λ *( A ) for any Lebesgue-measurable set A . The existence of sets that are not Lebesgue-measurable is a consequence of the set-theoretical axiom of choice , which is independent from many of the conventional systems of axioms for set theory . The Vitali theorem , which follows from the axiom, states that there exist subsets of R that are not Lebesgue-measurable. Assuming

2211-551: Is described below. An open interval does not include any endpoint, and is indicated with parentheses. For example, ( 0 , 1 ) = { x ∣ 0 < x < 1 } {\displaystyle (0,1)=\{x\mid 0<x<1\}} is the interval of all real numbers greater than 0 and less than 1 . (This interval can also be denoted by ]0, 1[ , see below). The open interval (0, +∞) consists of real numbers greater than 0 , i.e., positive real numbers. The open intervals are thus one of

2278-434: Is left-closed and right-open. The empty set and the set of all reals are both open and closed intervals, while the set of non-negative reals, is a closed interval that is right-open but not left-open. The open intervals are open sets of the real line in its standard topology , and form a base of the open sets. An interval is said to be left-closed if it has a minimum element or is left-unbounded, right-closed if it has

2345-529: Is not in E {\displaystyle E} : the set difference of A {\displaystyle A} and E {\displaystyle E} . These partitions of A {\displaystyle A} are subject to the outer measure. If for all possible such subsets A {\displaystyle A} of the real numbers, the partitions of A {\displaystyle A} cut apart by E {\displaystyle E} have outer measures whose sum

2412-405: Is said left-open or right-open depending on whether the excluded endpoint is on the left or on the right. These intervals are denoted by mixing notations for open and closed intervals. For example, (0, 1] means greater than 0 and less than or equal to 1 , while [0, 1) means greater than or equal to 0 and less than 1 . The half-open intervals have the form Every closed interval

2479-414: Is said to be left-bounded or right-bounded , if there is some real number that is, respectively, smaller than or larger than all its elements. An interval is said to be bounded , if it is both left- and right-bounded; and is said to be unbounded otherwise. Intervals that are bounded at only one end are said to be half-bounded . The empty set is bounded, and the set of all reals is the only interval that

2546-594: Is sometimes called an n {\displaystyle n} -dimensional interval . Carath%C3%A9odory%27s criterion Carathéodory's criterion is a result in measure theory that was formulated by Greek mathematician Constantin Carathéodory that characterizes when a set is Lebesgue measurable . Carathéodory's criterion : Let λ ∗ : P ( R n ) → [ 0 , ∞ ] {\displaystyle \lambda ^{*}:{\mathcal {P}}(\mathbb {R} ^{n})\to [0,\infty ]} denote

2613-442: Is tested by taking subsets A {\displaystyle A} of the real numbers using E {\displaystyle E} as an instrument to split A {\displaystyle A} into two partitions: the part of A {\displaystyle A} which intersects with E {\displaystyle E} and the remaining part of A {\displaystyle A} which

2680-740: Is the Cartesian product of n {\displaystyle n} finite intervals. For n = 2 {\displaystyle n=2} this is a rectangle ; for n = 3 {\displaystyle n=3} this is a rectangular cuboid (also called a " box "). Allowing for a mix of open, closed, and infinite endpoints, the Cartesian product of any n {\displaystyle n} intervals, I = I 1 × I 2 × ⋯ × I n {\displaystyle I=I_{1}\times I_{2}\times \cdots \times I_{n}}

2747-448: Is the complement of M . {\displaystyle M.} The family of all μ ∗ {\displaystyle \mu ^{*}} –measurable subsets is a σ-algebra (so for instance, the complement of a μ ∗ {\displaystyle \mu ^{*}} –measurable set is μ ∗ {\displaystyle \mu ^{*}} –measurable, and

Lebesgue measure - Misplaced Pages Continue

2814-404: Is the corresponding closed ball, and the interval's two endpoints { a , b } {\displaystyle \{a,b\}} form a 0-dimensional sphere . Generalized to n {\displaystyle n} -dimensional Euclidean space , a ball is the set of points whose distance from the center is less than the radius. In the 2-dimensional case, a ball is called a disk . If

2881-410: Is the outer measure of A {\displaystyle A} , then the outer Lebesgue measure of E {\displaystyle E} gives its Lebesgue measure. Intuitively, this condition means that the set E {\displaystyle E} must not have some curious properties which causes a discrepancy in the measure of another set when E {\displaystyle E}

2948-407: Is unbounded at both ends. Bounded intervals are also commonly known as finite intervals . Bounded intervals are bounded sets , in the sense that their diameter (which is equal to the absolute difference between the endpoints) is finite. The diameter may be called the length , width , measure , range , or size of the interval. The size of unbounded intervals is usually defined as +∞ , and

3015-422: Is used as a "mask" to "clip" that set, hinting at the existence of sets for which the Lebesgue outer measure does not give the Lebesgue measure. (Such sets are, in fact, not Lebesgue-measurable.) The Lebesgue measure on R has the following properties: All the above may be succinctly summarized as follows (although the last two assertions are non-trivially linked to the following): The Lebesgue measure also has

3082-484: Is viewed as a metric space , its open balls are the open bounded intervals  ( c  +  r ,  c  −  r ) , and its closed balls are the closed bounded intervals  [ c  +  r ,  c  −  r ] . In particular, the metric and order topologies in the real line coincide, which is the standard topology of the real line. Any element  x of an interval  I defines a partition of  I into three disjoint intervals I 1 ,  I 2 ,  I 3 : respectively,

3149-638: The Carathéodory criterion , which requires that for every A ⊆ R {\displaystyle A\subseteq \mathbb {R} } , The sets E {\displaystyle E} that satisfy the Carathéodory criterion are said to be Lebesgue-measurable, with its Lebesgue measure being defined as its Lebesgue outer measure: λ ( E ) = λ ∗ ( E ) {\displaystyle \lambda (E)=\lambda ^{\!*\!}(E)} . The set of all such E {\displaystyle E} forms

3216-541: The Euclidean metric on R (or any metric Lipschitz equivalent to it). On the other hand, a set may have topological dimension less than n and have positive n -dimensional Lebesgue measure. An example of this is the Smith–Volterra–Cantor set which has topological dimension 0 yet has positive 1-dimensional Lebesgue measure. In order to show that a given set A is Lebesgue-measurable, one usually tries to find

3283-566: The Lebesgue outer measure on R n , {\displaystyle \mathbb {R} ^{n},} where P ( R n ) {\displaystyle {\mathcal {P}}(\mathbb {R} ^{n})} denotes the power set of R n , {\displaystyle \mathbb {R} ^{n},} and let M ⊆ R n . {\displaystyle M\subseteq \mathbb {R} ^{n}.} Then M {\displaystyle M}

3350-458: The complement of M . {\displaystyle M.} Notice that S {\displaystyle S} is not required to be a measurable set. The Carathéodory criterion is of considerable importance because, in contrast to Lebesgue's original formulation of measurability, which relies on certain topological properties of R , {\displaystyle \mathbb {R} ,} this criterion readily generalizes to

3417-408: The coordinates of a point or vector in analytic geometry and linear algebra , or (sometimes) a complex number in algebra . That is why Bourbaki introduced the notation ] a , b [ to denote the open interval. The notation [ a , b ] too is occasionally used for ordered pairs, especially in computer science . Some authors such as Yves Tillé use ] a , b [ to denote the complement of

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3484-494: The endpoints of the interval. In countries where numbers are written with a decimal comma , a semicolon may be used as a separator to avoid ambiguity. To indicate that one of the endpoints is to be excluded from the set, the corresponding square bracket can be either replaced with a parenthesis, or reversed. Both notations are described in International standard ISO 31-11 . Thus, in set builder notation , Each interval (

3551-403: The greatest lower bound (infimum) of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit E {\displaystyle E} most tightly and do not overlap. That characterizes the Lebesgue outer measure. Whether this outer measure translates to the Lebesgue measure proper depends on an additional condition. This condition

3618-570: The intermediate value theorem asserts that the image of an interval by a continuous function is an interval; integrals of real functions are defined over an interval; etc. Interval arithmetic consists of computing with intervals instead of real numbers for providing a guaranteed enclosure of the result of a numerical computation, even in the presence of uncertainties of input data and rounding errors . Intervals are likewise defined on an arbitrary totally ordered set, such as integers or rational numbers . The notation of integer intervals

3685-403: The Lebesgue measure on those sets for which it is defined; however, there are many more Lebesgue-measurable sets than there are Borel measurable sets. The Borel measure is translation-invariant, but not complete . The Haar measure can be defined on any locally compact group and is a generalization of the Lebesgue measure ( R with addition is a locally compact group). The Hausdorff measure

3752-428: The above definitions and terminology. For instance, the interval (−∞, +∞)  =  R {\displaystyle \mathbb {R} } is closed in the realm of ordinary reals, but not in the realm of the extended reals. When a and b are integers , the notation ⟦ a, b ⟧, or [ a .. b ] or { a .. b } or just a .. b , is sometimes used to indicate the interval of all integers between

3819-464: The axiom of choice, non-measurable sets with many surprising properties have been demonstrated, such as those of the Banach–Tarski paradox . In 1970, Robert M. Solovay showed that the existence of sets that are not Lebesgue-measurable is not provable within the framework of Zermelo–Fraenkel set theory in the absence of the axiom of choice (see Solovay's model ). The Borel measure agrees with

3886-438: The elements of  I that are less than  x , the singleton  [ x , x ] = { x } , {\displaystyle [x,x]=\{x\},} and the elements that are greater than  x . The parts I 1 and I 3 are both non-empty (and have non-empty interiors), if and only if x is in the interior of  I . This is an interval version of the trichotomy principle . A dyadic interval

3953-450: The equality μ ∗ ( S ) = μ ∗ ( S ∩ M ) + μ ∗ ( S ∩ M c ) {\displaystyle \mu ^{*}(S)=\mu ^{*}(S\cap M)+\mu ^{*}\left(S\cap M^{c}\right)} holds where M c := Ω ∖ M {\displaystyle M^{c}:=\Omega \setminus M}

4020-412: The form [ a , b ] intervals and sets of the form ( a , b ) segments throughout. These terms tend to appear in older works; modern texts increasingly favor the term interval (qualified by open , closed , or half-open ), regardless of whether endpoints are included. The interval of numbers between a and b , including a and b , is often denoted [ a ,  b ] . The two numbers are called

4087-453: The forms where a {\displaystyle a} and b {\displaystyle b} are real numbers such that a ≤ b . {\displaystyle a\leq b.} When a = b {\displaystyle a=b} in the first case, the resulting interval is the empty set ( a , a ) = ∅ , {\displaystyle (a,a)=\varnothing ,} which

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4154-407: The image of an interval by any continuous function from R {\displaystyle \mathbb {R} } to R {\displaystyle \mathbb {R} } is also an interval. This is one formulation of the intermediate value theorem . The intervals are also the convex subsets of R . {\displaystyle \mathbb {R} .} The interval enclosure of

4221-405: The interval  ( a ,  b ) ; namely, the set of all real numbers that are either less than or equal to a , or greater than or equal to b . In some contexts, an interval may be defined as a subset of the extended real numbers , the set of all real numbers augmented with −∞ and +∞ . In this interpretation, the notations [−∞,  b ]  , (−∞,  b ]  , [ a , +∞]  , and [

4288-439: The interval. Dyadic intervals have the following properties: The dyadic intervals consequently have a structure that reflects that of an infinite binary tree . Dyadic intervals are relevant to several areas of numerical analysis, including adaptive mesh refinement , multigrid methods and wavelet analysis . Another way to represent such a structure is p-adic analysis (for p = 2 ). An open finite interval (

4355-400: The property of being σ -finite . A subset of R is a null set if, for every ε > 0, it can be covered with countably many products of n intervals whose total volume is at most ε. All countable sets are null sets. If a subset of R has Hausdorff dimension less than n then it is a null set with respect to n -dimensional Lebesgue measure. Here Hausdorff dimension is relative to

4422-575: The size of the empty interval may be defined as 0 (or left undefined). The centre ( midpoint ) of a bounded interval with endpoints a and b is ( a  +  b )/2 , and its radius is the half-length | a  −  b |/2 . These concepts are undefined for empty or unbounded intervals. An interval is said to be left-open if and only if it contains no minimum (an element that is smaller than all other elements); right-open if it contains no maximum ; and open if it contains neither. The interval [0, 1) = { x | 0 ≤ x < 1} , for example,

4489-423: The supremum does not exist, one says that the corresponding endpoint is + ∞ . {\displaystyle +\infty .} Intervals are completely determined by their endpoints and whether each endpoint belong to the interval. This is a consequence of the least-upper-bound property of the real numbers. This characterization is used to specify intervals by mean of interval notation , which

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