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In mathematics, a subset A of a Polish space X is universally measurable if it is measurable with respect to every complete probability measure on X that measures all Borel subsets of X. In particular, a universally measurable set of reals is necessarily Lebesgue measurable (see #Finiteness condition) below.

Every analytic set is universally measurable. It follows from projective determinacy, which in turn follows from sufficient large cardinals, that every projective set is universally measurable.

Reference:
http://en.wikipedia.org/wiki/Universally_measurable_set

In mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or volume to subsets of Euclidean space. It is used throughout real analysis, in particular to define Lebesgue integration. Sets which can be assigned a volume are called Lebesgue measurable; the volume or measure of the Lebesgue measurable set A is denoted by λ(A). A Lebesgue measure of ∞ is possible, but even so, assuming the axiom of choice, not all subsets of R^n are Lebesgue measurable. The "strange" behavior of non-measurable sets gives rise to such statements as the Banach-Tarski paradox, a consequence of the axiom of choice.

Lebesgue measure is often denoted dx, but this should not be confused with the distinct notion of a volume form.

Reference:
http://en.wikipedia.org/wiki/Lebesgue_measure

In topology and related fields of mathematics, a set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U. In other words, the distance between any point x in U and the edge of U is always greater than zero.

As an example, consider the open interval (0,1) consisting of all real numbers x with 0 < x < 1. Here, the topology is the usual topology on the real line. We can look at this in two ways. Since any point in the interval is different from 0 and 1, the distance from that point to the edge is always non-zero. Or equivalently, for any point in the interval we can move by a small enough amount in any direction without touching the edge and still be inside the set. Therefore, the interval (0,1) is open. However, the interval (0,1] consisting of all numbers x with 0 < x ≤ 1 is not open; if one takes x = 1 and moves even the tiniest bit in the positive direction, one will be outside of (0,1].

Reference:
http://en.wikipedia.org/wiki/Open_set

In mathematics, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians, like Sierpinski, Kuratowski, Tarski, and others. However, Polish spaces are primarily studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations.

Common examples of Polish spaces are the real line, the Cantor space, and Baire space.

Between any two uncountable Polish spaces, there is a Borel isomorphism; that is, a bijection that preserves the Borel structure. In particular, every uncountable Polish space has the cardinality of the continuum.

Lusin spaces, Suslin spaces, and Radon spaces are generalizations of Polish spaces.

Reference:
http://en.wikipedia.org/wiki/Polish_space

In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that
f(x) = y.

Alternatively, f is bijective if it is a one-to-one correspondence between those sets; i.e., both one-to-one (injective) and onto (surjective). (It should be noted that one-to-one function means one-to-one correspondence (i.e., bijection) to some authors, but injection to others.)

For example, consider the function succ, defined from the set of integers  to , that to each integer x associates the integer succ(x) = x + 1. For another example, consider the function sumdif that to each pair (x,y) of real numbers associates the pair sumdif(x,y) = (x + y, x − y).

A bijective function from a set to itself is also called a permutation.

The set of all bijections from X to Y is denoted as XY.

Bijective functions play a fundamental role in many areas of mathematics, for instance in the definition of isomorphism (and related concepts such as homeomorphism and diffeomorphism), permutation group, projective map, and many others.

Reference:
http://en.wikipedia.org/wiki/Bijection

In mathematics, an injective function is a function which associates distinct arguments with distinct values.

An injective function is called an injection, and is also said to be an information-preserving or one-to-one function (the latter is not to be confused with one-to-one correspondence, i.e. a bijective function).

A function f that is not injective is sometimes called many-to-one. (However, this terminology is also sometimes used to mean "single-valued", i.e. each argument is mapped to at most one value.)

Reference:
http://en.wikipedia.org/wiki/Injective_function

In mathematics, a function f is said to be surjective if its values span its whole codomain; that is, for every y in the codomain, there is at least one x in the domain such that f(x) = y .

Said another way, a function f: X → Y is surjective if and only if its range f(X) is equal to its codomain Y. A surjective function is called a surjection, and also said to be onto.

Reference:
http://en.wikipedia.org/wiki/Surjective_function

In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M.

Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because √2 is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it. (See the examples below.) It is always possible to "fill all the holes", leading to the completion of a given space, as will be explained below.

Reference:
http://en.wikipedia.org/wiki/Complete_space

In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined.

The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space. In fact, the notion of "metric" is a generalization of the Euclidean metric arising from the four long known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line connecting them.

The geometric properties of the space depends on the metric chosen, and by using a different metric we can construct interesting non-Euclidean geometries such as those used in the theory of general relativity.

A metric space also induces topological properties like open and closed sets which leads to the study of even more abstract topological spaces.

Reference:
http://en.wikipedia.org/wiki/Metric_space