[size=18.6667px]数列上下极限与集列上下极限集的定义比较
[size=18.6667px]文/王羹渊
我们在高等数学中学过,单调有界数列必有极限,而且极限就是其确界,这里的极限是指有限数,是所谓的正常极限。如果一个数列单调且无界,则其极限是无穷大,这是所谓的广义极限。比如单调递增无界数列有极限为正无穷大,单调递减数列有极限为负无穷大。这与单调有界数列必有极限合起来,就是单调数列必有极限。
同样,在一个样本空间中,单调集列必有极限,比如设AnìAn+1,那么{An}是一个单调递增的集列,这个集列的极限定义为LimAn=∪An,其中并集是从n=1到n=∞求并运算。同样,单调递减集列{An}满足 AnéAn+1,其极限定义为LimAn=∩An,其中交集是从n=1到n=∞求交运算。
单调数列或单调集列必有极限的原则可以被用来定义任意数列或集列的上极限与下极限。我们存在以下的类比。
Let {Xn} is a sequence of number, we defineYn=sup{Xn,Xn+1,…}, then the sequence {Yn} is monotonicallydecreasing. But we should have in mind that {Xn} need not be decreasing. Adecreasing sequence has a limit, which may be negative infinity, and the limitmust be its infimum. So the inf{Yn} exists and is just the limit of {Yn}. Thelimit of {Yn} is defined as the upper limit of {Xn}, which is denoted by:
For the sequence of number {Xn}, we define Zn=inf{Xn,Xn+1,…}, then the sequence {Zn} is monotonically increasing. Andan increasing sequence of number must have a limit, and the limit must equalthe supermum of the sequence. So sup{Zn} exists and we define it as the lowerlimit of {Xn}.
For the set {Xn,Xn+1,…}, the supremum must be always greater thanthe infimum, i.e.,
Yn=sup{Xn,Xn+1,…}3 Zn=inf{Xn,Xn+1,…}for all nÎN
So Lim{Yn}3Lim{Zn}, but Lim{Yn}=Inf{Yn} andLim{zn}=sup{Zn}, so we have inf{Yn}3sup{Zn}, i.e.,
3
让{An}为任意集列,定义Bn=
让{An}为任意集列,定义Cn=
数列的上下极限与集列的上下极限,在定义的结构上非常类似。下表具体比较了数列上下极限与集列上下极限集的定义。
|
| 数列{Xn} | 集列{An} |
上极限 | 过渡数列或集列的定义 | Yn= sup{Xn,Xn+1,…} | Bn= |
过渡数列或集列的性质 | {Yn}为递减数列 | {Bn}为递减集列 | |
过渡数列或集列的极限 | LimYn=inf{Yn} | LimBn=∩{Bn} | |
原数列或集列的上极限 | |||
下极限 | 过渡数列或集列的定义 | Zn=inf{Xn,Xn+1,…} | Cn= |
过渡数列或集列的性质 | {Zn}为递增数列 | {Cn}为递增集列 | |
过渡数列或集列的极限 | LimZn=sup{Yn} | LimCn=∪{Cn} | |
原数列或集列的上极限 |