在這篇論文，我們主要的目的是要研究關於弱幾乎收斂序列以σ-極限的觀點來呈現的一些基本性質。在1996年，李源泉與蕭勝彥已證明σ-極限與弱幾乎收斂是等價的觀點。
弱幾乎收斂已經被很多數學家應用到nonexpansive函數的定點定理上，如Baillon，Bruck，Reich，Hirano，Brezis及Browder等等。
在1975年，Baillon證明了下述的非線性ergodic定理：令C是一個Hilbert空間X的封閉凸子集。若T是從C對應到C的nonexpansive函數，即對所有在C內的元素u,v，T滿足
‖Tu-Tv‖ ‖u-v‖，有一個定點C內的元素y，則對所有C內的元素x，{ Tnx }會弱幾乎收斂到y。一般來說，即使X是一個Hilber空間，{ Tnx }也不會弱幾乎收斂到一個定點。Bruc和Reich把這個結果推展到X是均勻凸Banach空間有Frechet可微norm及不同的充分條件。另一方面，Baillon也證明了下列的強ergodic定理：若X是一個Hilbert空間，-C = C，及T是奇函數，則對每個C內的元素x，{ Tnx }會強幾乎收斂到T的定點y，即
∥ Tk+mx-y∥= 0 均勻在m 0上。
Brezis及Browder證明即使T函數的條件消弱如下，則Baillon的結果仍然是對的：
0是C內的元素，對C內的元素u,v
‖Tu+Tv‖ ‖u+v‖2+c[‖u‖2-‖Tu‖2+‖u‖2-‖Tv‖2] (1.1)
其中c 是一個非負的常數。我們知道如果T是在Hilbert空間X的一個閉凸子集C上的一個nonexpansive函數，而且滿足(1.1)式，則T會滿足：
對所有C內元素u,v，
∥Tn+iu- Tnv∥存在，均勻在i 0上。 (1.2)
Bruck證明如果C是一個Hilbert空間X上的一個閉凸子集，T是一個滿足(1.2)式的nonexpansive函數且有一個定點，則對每個C內的元素x，{ Tnx }會強幾乎收斂到T的定點。不久，Kobayasi及Miyadera證明即使X是一個均勻凸Banach空間，Bruck的結果仍然是對的。Hirano，Takahashi及Oka證明上述T的條件可以消弱如下：
∥Tku- Tkv∥ ak‖u-v‖ 對所有C內的元素u,v及k 0 (1.3)
其中ak 是一個滿足 ak =1的非負常數。在這種情形，T被稱為asymptotically nonexpansive。若T是從C對應到C的asymptotically nonexpansive奇函數，則它會滿足
∥Tku+ Tkv∥ ak‖u+v‖ 對所有C內的元素u,v及k 0 (1.4)
其中ak 是一個滿足 ak =1的非負常數。
在這篇論文的第二節，我們證明了如果N是一個實Banach空間X 的proper closed cone，f是從N對應到N的函數，在0點弱連續且滿足f(0)=0，則對每個N內的序列{ x n}，滿足{ x n}的σ-極限為0且{f ( x n)}是有界的，則{f ( x n)} 的σ-極限為0。
已知若(Ω,Σ,μ)是一個測度空間，且對所有的n=1，2，3…，fn是從Ω對應到複數值的Lebesgue可測函數，使得 fn =f a.e. 則f是可測函數。由弱幾乎收斂的定義可知，如果σ-lim fn=f a.e. 則f也是可測函數。在第三節，我們提供了一個和dominated收斂定理等價的另一個定理，敘述如下：假設(Ω,Σ,μ)是一個測度空間，且g，f，f1，f2，…都是從Ω對應到複數值的的可測函數。對所有的n=1,2,3…,
｜fn｜≦g a.e. ÎL1(μ) 而且σ-lim fn (ω)=f (ω) a.e.[μ]，則f是μ-可積且 = = 。
我們很容易看出弱幾乎收斂的條件比弱收斂還弱，所以若f在點x上弱連續，則{ x n}的σ-極限為x不一定會導致{ f (x n)}的σ-極限為f (x)。在定理3.5我們證明了若x是一個複數，f是複數值函數，則f在x連續若且惟若對所有複數序列{ x n}，{｜x n -x｜}的σ-極限為0可推至{f ( x n)}的σ-極限為f (x)。從定理3.6到序理3.9，我們研究在怎樣的充分條件下，純數x及有界數列{ x n }滿足{ x n}的σ-極限為x可以推至{ f (x n)}的σ-極限為f (x)。例如，若a為實數且{ a n }是實數有界數列，滿足{ a n }的σ-極限為a，且對所有的n 1，a n a都成立，則對所有的p=1,2…, an p = a p都成立。最後，在第四節我們給了兩個例子。

In this paper, our primary objective is to study basic poroperties about
weakly almost-convergent squence in terms of the conception of σ-limits. In 1996, Li and Shaw [11] showed that the conception ofσ-limit is equivalent to the weak almost-convergence (see Definition 2.3).
The weak almost-convergence had been applied to the fixed point theory of nonexpansive mappings by many mathematicians, for example, Baillon[1], Bruck[3,4], Reich, Hirano[7], Brezis and Browder[2], etc.
In section 2, we show that if N is a proper closed cone of a real Banach space X and if f:N->N is weak-weak continuous at 0 with f(0)=0, then for every sequence {xn} in N such thatσ-lim xn=0 and {f(xn)} is bounded implyσ-lim f(xn)=0.(see Proposition 2.9)
It is well known that if (Ω,Σ,μ) is a measure space and
fn:Ω->C, for n=1,2,… , are Lebesgue measurable functions such that limn fn=f a.e. then f is measurable. By the definition of weakly almost-convergence, f is also measurable if σ-lim fn = f a.e. [μ]. In section 3, we give another version of the dominated convergence theorem stated as following: Suppose (Ω,Σ,μ) is a measure space and g,f,f1, f2,… : Ω->C are measurable.
Suppose fn≦ g (a.e.) in L1 (μ) for all n=1,2,… and
σ-lim fn (ω) = f(ω) a.e. [μ].
Then f is integrable .
It is easy to see that the weakly almost convergence is weaker than the weak convergence .
From Proposition 3.7 to Corollary 3.10, we study under which sufficient conditions at a scalar x and a bounded sequece { xn } with σ-lim xn = x we have
σ-lim f(xn) = f(x).
Finally, we give two examples in section 4.

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