${\mu}$- Integrable Functions and Weak Convergence of Finite Measures

This paper deals with functions that defined in metric spaces and valued in complete paranormed vector spaces or valued in Banach spaces, and obtains some necessary and sufficient conditions for weak convergence of finite measures.

Abstract This paper deals with functions that defined in metric spaces and valued in complete paranormed vector spaces or valued in Banach spaces, and obtains some necessary and sufficient conditions for weak convergence of finite measures. Our main results are as follows. Let X be a Banach space, ) , , and proved that the function g is not necessarily to be point-wise continuous. Wei [3] worked with functions valued in a metric space X, and assumed that } { n µ is tight and weak convergence of the finite In this paper, we extended the necessary and sufficient condition to any Banach space, and to any Frechet space (in Bourbaki's terminology) that has a base. We generalized Nielsen's result in [1], and generalized the corresponding results of Yang and Wei in [2,3].
Let K be the field of real numbers or the field of complex numbers, and X be a vector space over K. A paranormed space is a pair (X, || || ⋅ ), where || || ⋅ is a function, called a paranorm, such that In what follows paranormed paces will always be regarded as metric spaces with respect to the metric || || ⋅ .
It is known that a normed vector space is a paranormed vector space, but a paranormed space is not necessary to be a normed vector space.
Any Banach space is a complete paranormed space. But the converse is not true.
A complete paranormed space is called a Frechet space in Bourbaki's terminology, see [4].
Let Ω be an arbitrary non-empty set. A family Σ of subsets of Ω is called ring if (a) Σ contains the empty set Φ ; It follows that also the intersection A ∩ B belongs to Σ because is also in Σ. Hence, a ring is closed under taking the set-theoretic operations ∩ , ∪ , \.
of sets from Σ is also in Σ. A σ-ring in Ω that contains Ω is called a σ-algebra of Ω.
Let Ω be an arbitrary non-empty set and Σ be a σ-algebra of Ω, R+ the set of all nonnegative real numbers. A functional µ: Σ → R+ called a σ-additive measure on Σ if whenever a set A ∈ Σ is a disjoint union of an at most countable sequence in Σ, i.e., A = If N = ∞ then the above sum is understood as a series.
(Ω, Σ, µ) is called a finite measure space if the measure µ on Σ is finite.
The µ-Integral of the simple function g is defined as i.e., there exists A∈ Σ and 0 ( In this case, exists, and the µ-Integral of g is defined as When X is a Banach space, the µ-Integral is known as Bochner integral [5,6]. Where n x' are called the coordinate functionals (n = 1, 2, …).

Some Lemmas
In what follows, let ) , Lemma 1 can be found in [7].

Lemma 1 If a complete paranormed space X has a base {xn}, then {xn} is a Schauder base.
A seminorm is map for any scalar α .
From [4] one has the following Lemma 2.
Lemma 2 X is a complete paranormed space if and only if there is a family of continuous siminorms P = {pn; n = 1, 2, ….} on X, such that And the paranorm on X can be given by Furthermore, for any topological net , and X x ∈ , the following are equivalent A set X A ⊂ is called separable if A has a countable dense subset, i.e., there exists a countable subset For X A ⊂ , span A is the set of all possible linear combinations of the elements in A.

Lemma 3 Suppose
Π ∈ µ . A µ-measurable function g: Ω → X is µ-integrable if and only if (a) g is µ-essential separable valued, i.e., there exists where P is defined as in Lemma 2.
The following Lemma 4 is from [8].

Lemma 7
If g: Ω → X is a continuous function, then g is µ-integrable.

Weak Convergence of Finite Measures
The following is the traditional definition of weak convergence of measures [10].
This paper extends the concept of weakly convergent measures to complete paranormed spaces. If X is a separable Banach space, then X has a Schauder base. Similar to the proof of Theorem 1 one has the following Corollary 3. .

Corollary 3 Let
Let Ω ∈ n ς be random elements (n
This paper obtains some necessary and sufficient conditions for the weak convergence of finite measures in complete paranormed vector spaces that have bases and in general Banach spaces. Or, actually, we extend the concept of weak convergence of probability measures to complete Paranormed vector spaces that have bases, and to general Banach spaces.
We generalize also the concept of convergence of random variables in probability distributions, to Paranormed vector spaces and to general Banach spaces.