Displaying similar documents to “Kolmogorov's Zero-One Law”

The Vector Space of Subsets of a Set Based on Symmetric Difference

Jesse Alama (2008)

Formalized Mathematics

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For each set X, the power set of X forms a vector space over the field Z2 (the two-element field {0, 1} with addition and multiplication done modulo 2): vector addition is disjoint union, and scalar multiplication is defined by the two equations (1 · x:= x, 0 · x := ∅ for subsets x of X). See [10], Exercise 2.K, for more information.MML identifier: BSPACE, version: 7.8.05 4.89.993

Borel-Cantelli Lemma

Peter Jaeger (2011)

Formalized Mathematics

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This article is about the Borel-Cantelli Lemma in probability theory. Necessary definitions and theorems are given in [10] and [7].

On L 1 Space Formed by Real-Valued Partial Functions

Yasushige Watase, Noboru Endou, Yasunari Shidama (2008)

Formalized Mathematics

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This article contains some definitions and properties refering to function spaces formed by partial functions defined over a measurable space. We formalized a function space, the so-called L1 space and proved that the space turns out to be a normed space. The formalization of a real function space was given in [16]. The set of all function forms additive group. Here addition is defined by point-wise addition of two functions. However it is not true for partial functions. The set of partial...

On systems of null sets

K. Bhaskara Rao, R. Shortt (1999)

Colloquium Mathematicae

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The collection of all sets of measure zero for a finitely additive, group-valued measure is studied and characterised from a combinatorial viewpoint.

On L p Space Formed by Real-Valued Partial Functions

Yasushige Watase, Noboru Endou, Yasunari Shidama (2010)

Formalized Mathematics

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This article is the continuation of [31]. We define the set of Lp integrable functions - the set of all partial functions whose absolute value raised to the p-th power is integrable. We show that Lp integrable functions form the Lp space. We also prove Minkowski's inequality, Hölder's inequality and that Lp space is Banach space ([15], [27]).

The Friendship Theorem

Karol Pąk (2012)

Formalized Mathematics

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In this article we prove the friendship theorem according to the article [1], which states that if a group of people has the property that any pair of persons have exactly one common friend, then there is a universal friend, i.e. a person who is a friend of every other person in the group

Explicit construction of integral bases of radical function fields

Qingquan Wu (2010)

Journal de Théorie des Nombres de Bordeaux

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We give an explicit construction of an integral basis for a radical function field K = k ( t , ρ ) , where ρ n = D ∈ k [ t ] , under the assumptions [ K : k ( t ) ] = n and c h a r ( k ) ∤ n . The field discriminant of K is also computed. We explain why these questions are substantially easier than the corresponding ones in number fields. Some formulae for the P -signatures of a radical function field are also discussed in this paper.