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lower bound – non-denumerable
lower bound
A lower bound of an ordered set is an element which is less than or equal to every element in the set.
 Cf. greatest lower bound.
l.u.b
Abbreviation for least upper bound.
Mahlo cardinal
A cardinal k is called Mahlo if it is inaccessible and {a < k:a is inaccessible} is stationary in k.
maximal filter
See filter.
measurable cardinal
If there exists a non-principal k-complete ultrafilter on a set of size k, then k is called a measurable cardinal.
measurable function
Given measure spaces (X, M, m) and (Y, N, n), a function f from X to Y is said to be measurable if the inverse image of every measurable set in Y is measurable in X.
measurable space
A set X together with a s-algebra of sets M defined on it is called a measurable space, usually denoted (X, M). The elements of M are called measurable sets.
Cf. measure.
measure
Given a set X and a s-algebra of sets M defined on X, a measure on X is a positive real-valued function m whose domain is M, and which satisfies: - the measure of the empty set is zero.
- (Countable additivity) For any countable, disjoint sequence of sets in the domain of m, the measure of the union of the sequence is equal to the sum of the measures of the sets in the sequence.
If the measure of every set in M is finite, then m is called a finite measure. If X can be expressed as a union of sets in M all of whose measures are finite, then m is called a s-finite measure. If for every set E in the domain of m whose measure is infinite there is a subset F of E which has finite measure, then m is called a semi-finite measure. If for any set E in M which has zero measure all the subsets of E are also in M (and therefore have zero measure), then m is called a complete measure.
Cf. measure space, signed measure, outer measure, Lebesgue measure, Borel measure.
measure space
A set X together with a s-algebra of sets M defined on it and a measure m defined on M is called a measure space, usually denoted by (X, M, m).
meet
A binary operation whose value on two elements a and b of a lattice is the greatest lower bound of a and b, denoted a  b.
Cf. join.
meet-morphism
A function f on a lattice L is called a meet-morphism if for every a and b in L we have f(a b) = f(a) f(b). If f has an inverse that is also a meet-morphism, then f is called a meet-isomorphism. A meet-isomorphism from a lattice to itself is called a meet-automorphism.
Cf. meet, join-morphism.
metric
Given a set X, a metric on X is a function d with domain X and range in R* (the set of non-negative real numbers) satisfying, for all a, b, and c in X:- d(a,b) = 0 if and only if a = b;
- d(a,b) = d(b,a); and
- d(a,b) + d(b,c) >= d(a,c
(where >= means ‘greater than or equal to’).
Cf. metric space.
metric space
A set with a metric defined on its elements.
monotone function
Also called monotonic function. See order-preserving function.
morphism
A function from one set to another is called a morphism if it preserves some designated structural properties or operations on the domain set. Typically, the word morphism is not used by itself, but in combination with a prefix that indicates whether it is injective, surjective, etc.
Cf. automorphism, epimorphism, homeomorphsim, homomorphism, isomorphism, monomorphsim.
mutually singular
Given a measurable space (X, M), two signed measures m and n on X are called mutually singular, denoted by
if there exist sets E and F in M such that their union is X, their intersection is empty, and m(E) = 0 and n(F) = 0.
natural number
An element of the set N = {1, 2, 3, ...} consisting of all the “counting numbers.” When the number 0 is included, this set is sometimes called the whole numbers. In set theory, the natural numbers (incuding 0) are identified with the set w of finite ordinals. The natural numbers are a well-founded linear order with no largest member, and are countably infinite.
Cf. Peano axioms, rational number, real number.
Related MiniText: Number -- What Is How Many?
negation
If j is a statement, sentence, or formula of logic, then the negation of j, denoted by  j, is that formula which is true whenever j is false, and false whenever j is true.
negative set
Given a signed measure m on a measure space X, a measurable set A in X is called a negative set if the measure of all measurable subsets of A is less than or equal to zero.
Cf. positive set, null set.
neighborhood
A neighborhood of a point x of a topological space is an open set of the space containing x. In a metric space, a d-neighborhood of x is the collection of all points of the space whose distance from x is less than d.
non-denumerable
Uncountable.
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