Autocontinuity from below of set functions and convergence in measure

Jun Li, Masami Yasuda and Ling Zhou

Abstract In this note, the concepts of strong autocontinuity from below and strong converse autocontinuity from below of set function are introduced. By using four types of autocontinuity from below of monotone measure, the relationship between convergence in measure and pseudo-convergence in measure for sequence of measurable function are discussed.

Key words: Monotone measure, Autocontinuity from below, Convergence in measure, Pseudo-convergence in measure.

1 Introduction

In non-additive measure theory, there are several different kinds of convergence for sequence of measurable functions, such as almost everywhere convergence, pseudo-almost everywhere convergence, convergence in measure, and convergence pseudo-in measure. The implication relationship between such convergence concepts are closely related to the structural characteristics of set functions. In this direction there are a lot of results ([2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15]).

Jun Li School of Science, Communication University of China, Beijing 100024, China e-mail: lijun@cuc.edu.cn Masami Yasuda Department of Mathematics & Informatics, Faculty of Science, Chiba University, Chiba 263-8522, Japan e-mail: yasuda@math.s.chiba-u.ac.jp Ling Zhou College of Applied Sciences, Xinjiang Agricultural University, Xinjiang, Urumqi 830052, China e-mail: zhoulin5703@163.com

In this note, we further discuss the relationship between convergence in measure and convergence pseudo-in measure for sequence of measurable functions. We shall introduce the concepts of strong autocontinuity from below and strong converse autocontinuity from below of a set function. By using the two types of autocontinuity from below of monotone measures, we investigate the inheriting of convergence in measure and convergence pseudo-in measure for sequence of measurable function under the common addition operation “+” and logic addition operation “∨”. The implication relationship between convergence in measure and pseudo-convergence in measure are shown by using autocontinuity from below and converse autocontinuity from below, respectively.

2 Preliminaries

Let X be a non-empty set, F a σ-algebra of subsets of X, and (X, F) denotes the measurable space.

Definition 1. ([10, 15]) Set function μ : F → [0, +∞] is called a monotone measure on (X, F) iff it satisfies the following requirements:

(1) μ(∅) = 0; (vanishing at ∅)

(2) A ⊂ B and A, B ∈ F ⇒ μ(A) ≤ μ(B). (monotonicity)

When μ is a monotone measure, the triple (X, F, μ) is called a monotone measure space ([10, 15]).

In some literature, a set function μ satisfying the conditions (1) and (2) of Definition 1 is called a fuzzy measure or a non-additive measure .

In this paper, all the considered sets are supposed to belong to F and μ is supposed to be a finite monotone measure, i.e., μ(X) < ∞. All concepts and symbols not defined may be found in [10, 15].

Definition 2. ([1]) A set function μ : F → [0, +∞) is said to have pseudometric generating property (for short p.g.p), if for any {E_{n}} ⊂ F and {F_{n}} ⊂ F,

μ(E_{n}) ∨ μ(F_{n}) → 0 ⟹ μ(E_{n} ∪ F_{n}) → 0.

Note: The concept of pseudometric generated property goes back to Dobrakov and Farkova in seventies, and this was related to Frechet-Nikodym topology [1, 10].

Let F be the class of all finite real-valued measurable functions on (X, F, μ), and let A ∈ F, f ∈ F, f_{n} ∈ F (n = 1, 2, …) and {f_{n}} denote a sequence of measurable functions. We say that {f_{n}} converges in measure μ to f on A, and denote it by f_{n} ⟶_{A}^{μ} f, if for any given σ > 0, lim_{n→+∞}μ({|f_{n} −f| ≥

σ} ∩ A) = 0; {f_{n}} converges pseudo-in measure μ to f on A, and denote it

by f_{n} ⟶_{A}^{p, μ} f, if for any given σ > 0, lim_{n→+∞}μ({|f_{n} − f| < σ} ∩ A) = μ(A); {f_{n}}

converges pseudo-in measure μ to f in A, and denote it by f_{n} ⟶^{p.μ} f in A, if

f_{n} ⟶^{p.μ} f on C for all C ∈ A ∩ F.

3 Autocontinuity of set function

In [14] Wang introduced the concepts of autocontinuity from below and converse-autocontinuity from below of set function, and discussed the convergence for sequence of measurable functions by using the structure of set functions. Now we shall introduce the concepts of strong autocontinuity from below and strong converse-autocontinuity from below for set functions and show their properties.

Definition 3. ([10, 14, 15]) Let (X, F, μ) be a monotone measure space.

(1) μ is said to be autocontinuous from below and denote it by autoc.↑, if for any E ∈ F, {F_{n}} ⊂ F,

μ(F_{n}) → 0 ⟹ μ(E − F_{n}) → μ(E);

(2) μ is said to be converse-autocontinuous from below and denote it by c.autoc.↑, if for any A ∈ F, {B_{n}} ⊂ A ∩ F,

μ(B_{n}) → μ(A) ⟹ μ(A − B_{n}) → 0.

Definition 4. Let (X, F, μ) be a monotone measure space.

(1) μ is said to be strong autocontinuous from below and denote it by s.autoc.↑, if

μ(E_{n}) ∨ μ(F_{n}) → 0 ⟹ μ(A − E_{n} ∪ F_{n}) → μ(A),

for any A ∈ F, {E_{n}} ⊂ F and {F_{n}} ⊂ F;

(2) μ is said to be strong converse-autocontinuous from below and denote it by s.c.autoc.↑, if

μ(A − E_{n}) ∧ μ(A − F_{n}) → μ(A) ⟹ μ(E_{n} ∪ F_{n}) → 0,

for any A ∈ F, {E_{n}} ⊂ A ∩ F and {F_{n}} ⊂ A ∩ F.

Proposition 1. If μ is s.autoc.↑ (resp. s.c.autoc.↑), then it is autoc.↑ (resp. c.autoc.↑).

Proposition 2. If μ is autoc.↑ and has p.g.p, then it is s.autoc.↑.

Proposition 3. If μ is c.autoc.↑ and has p.g.p, then it is s.c.autoc.↑.

4 Convergence in measure

In this section, we study the application relationship between convergence in measure and convergence pseudo-in measure on monotone measure spaces.

The first conclusion of the following theorem due to Wang [15]. Theorem 1. Let μ be a monotone measure. Then,

(1) μ is autoc.↑ iff f_{n} ⟶_{A}^{p.μ} f whenever f_{n} ⟶_{A}^{μ} f, ∀A ∈ F, f, f_{n} ∈ F;

(2) μ is c.autoc↑, iff f_{n} ⟶_{A}^{μ} f whenever f_{n} ⟶_{A}^{p.μ} f, ∀A ∈ F, f, f_{n} ∈ F.

Proof. We only prove (2). Let μ be c.autoco↑. If f_{n} ⟶_{A}^{p.μ} f, then for any given σ > 0, we have

lim_{n→+∞}μ({|f_{n} − f| < σ} ∩ A) = μ(A)

and therefore, using the converse-autocontinuity from below of μ, we have

lim_{n→+∞}μ({|f_{n} − f| ≥ σ} ∩ A) = lim_{n→+∞}μ(A − {|f_{n} − f| < σ})
= 0.

So f_{n} ⟶_{A}^{μ} f.

Conversly, for any A ∈ F, {B_{n}} ⊂ A ∩ F, and μ(B_{n}) → μ(A), we define measurable function sequences {f_{n}} by

f_{n} = χ_{B_{n}} = \{ \begin{matrix}0 & if x ∉ B_{n} \\ 1 & if x ∈ B_{n}, \end{matrix}

n = 1, 2,..., and denote f ≡ 1. It is easy to see that f_{n} ⟶_{A}^{p.μ} f. If it implies f_{n} ⟶_{A}^{μ} f, then for σ = \frac{1}{2}, we have

lim_{n→+∞}μ(\{|f_{n} − f| ≥ \frac{1}{2}\} ∩ A) = 0.

As

\{|f_{n} − f| ≥ \frac{1}{2}\} ∩ A = \{1 − χ_{B_{n}} ≥ \frac{1}{2}\} ∩ A = A − B_{n}.

So lim_{n→+∞}μ(A − B_{n}) = 0. This shows that μ is c.autoc↑. ⊓ ⊔

The following theorems describe the inheriting of convergence in measure and convergence pseudo-in measure for sequence of measurable function under the common addition operation.

Theorem 2. Let μ be a monotone measure.

(1) If μ is s.autoc.↑, then f_{n} ⟶^{μ} f and g_{n} ⟶^{μ} g on A imply

αf_{n} + βg_{n} ⟶_{D}^{p.μ} αf + βg,

for any D ∈ A ∩ F, α, β ∈ R^{1}.

(2) If μ is s.c.autoc.↑, then f_{n} ⟶_{A}^{p.μ} f and g_{n} ⟶_{A}^{p.μ} g imply

αf_{n} + βg_{n} ⟶_{A}^{μ} αf + βg,

for any A ∈ F, α, β ∈ R^{1}.

Proof. It is similar to the proof of Theorem 1. ⊓ ⊔

The following Theorem 3 and 4 describe respectively the characteristics of strong autocontinuity from below and strong converse-autocontinuity from below of set functions.

Theorem 3. The following statements are equivalent:

(1) μ is s.autoc.↑;

(2) f_{n} + g_{n} ⟶_{A}^{p.μ} 0 whenever f_{n} ⟶_{A}^{μ} 0 and g_{n} ⟶_{A}^{μ} 0, ∀A ∈ F;

(3) f_{n} ∨ g_{n} ⟶_{A}^{p.μ} 0 whenever f_{n} ⟶_{A}^{μ} 0 and g_{n} ⟶_{A}^{μ} 0, ∀A ∈ F.

Proof. (1) ⟹ (2). It follows directly from Theorem 2 above.

(2) ⟹ (3). For any A ∈ F, if f_{n} ⟶_{A}^{μ} 0 and g_{n} ⟶_{A}^{μ} 0, then |f_{n}| ⟶_{A}^{μ} 0 and

|g_{n}| ⟶_{A}^{μ} 0. By condition (2), we have |f_{n}| + |g_{n}| ⟶^{p.μ} 0 on A, therefore, for any

σ > 0,

lim_{n→+∞}μ({|f_{n}| + |g_{n}| < σ} ∩ A) = μ(A).

Noting that |f_{n} ∨ g_{n}| ≤ |f_{n}| + |g_{n}|, we get

{|f_{n}| + |g_{n}| < σ} ∩ A ⊆ {|f_{n} ∨ g_{n}| < σ} ∩ A ⊆ A.

So

lim_{n→+∞}μ({|f_{n} ∨ g_{n}| < σ} ∩ A) = μ(A).

This shows f_{n} ∨ g_{n} ⟶^{p.μ} 0 on A.

(3)⟹(1). For any {E_{n}} ⊂ F, {F_{n}} ⊂ F with lim_{n→∞}μ(E_{n}) ∨ μ(F_{n}) = 0, we define measurable function sequences {f_{n}} and {g_{n}} by

f_{n} = χ_{E_{n}} = \{ \begin{matrix}0 & if x ∉ E_{n} \\ 1 & if x ∈ E_{n} \end{matrix}

and

g_{n} = χ_{F_{n}} = \{ \begin{matrix}0 & if x ∉ F_{n} \\ 1 & if x ∈ F_{n}, \end{matrix}

n = 1, 2,..., then f_{n} ⟶^{μ} 0 on A and g_{n} ⟶^{μ} 0 on A. Thus, f_{n} ∨ g_{n} ⟶^{μ} 0 on A. Therefore for σ = \frac{1}{2}, we have

lim_{n→+∞}μ(\{f_{n} ∨ g_{n} < \frac{1}{2}\} ∩ A) = μ(A).

Noting f_{n} ∨ g_{n} = χ_{E_{n}} ∨ χ_{F_{n}} = χ_{E_{n}∪F_{n}}, and

\{χ_{E_{n}} ∨ χ_{F_{n}} < \frac{1}{2}\} ∩ A = A − \{χ_{E_{n}} ∨ χ_{F_{n}} ≥ \frac{1}{2}\} = A − E_{n} ∪ F_{n}.

So

lim_{n→+∞}μ(A − E_{n} ∪ F_{n}) = μ(A).

That is, μ is s.autoc.↑. ⊓ ⊔

Theorem 4. The following statements are equivalent:

(1) μ is s.c.autoc.↑;

(2) f_{n} + g_{n} ⟶_{A}^{μ} 0 whenever f_{n} ⟶_{A}^{p.μ} 0 and g_{n} ⟶_{A}^{p.μ} 0, ∀A ∈ F;

(3) f_{n} ∨ g_{n} ⟶_{A}^{μ} 0 whenever f_{n} ⟶_{A}^{p.μ} 0 and g_{n} ⟶_{A}^{p.μ} 0, ∀A ∈ F.

Proof. It is similar to the proof of Theorem 3. ⊓ ⊔

Acknowledgements The first author was supported by NSFC Grant No.70771010. The second author was supported by JSPS Grant No.22540112.

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