z****e
发帖数: 2024 | 1 HI guys,
could some one explain the following confusion i have? Thanks.
1. since a sigma-algebra is also an algebra,
if subset collection "C" generates a sigma-algebra "a", then "a" is also the algebra generated by "C", right? I'm not sure.
2. Is this "a" unique? In case it is an algebra. Or in case it is a sigma-algebra? I think "a" is unique since it is the "smallest". But then "C" generates the same algebra and sigma-algebra, right?
3. if "a" is an algebra generated by "C", does "a" and "C" g | a***s
发帖数: 616 | 2
the algebra generated by "C", right? I'm not sure.
No. The sigma-algebra generated by "C" is larger than the algebra generated
by
"C". You can say "a" is an algebra containing "C".
algebra? I think "a" is unique since it is the "smallest". But then "C"
generates the same algebra and sigma-algebra, right?
It's the unique sigma-algebra.
same algebra? is this generated algebra still "a"? does "a" and "C" generate
the same sigma-algebra? is this generated sigma-algebra contained by "a"?
The algebra
【在 z****e 的大作中提到】
: HI guys,
: could some one explain the following confusion i have? Thanks.
: 1. since a sigma-algebra is also an algebra,
: if subset collection "C" generates a sigma-algebra "a", then "a" is also the algebra generated by "C", right? I'm not sure.
: 2. Is this "a" unique? In case it is an algebra. Or in case it is a sigma-algebra? I think "a" is unique since it is the "smallest". But then "C" generates the same algebra and sigma-algebra, right?
: 3. if "a" is an algebra generated by "C", does "a" and "C" g
| z****e
发帖数: 2024 | 3 多谢。
赞。
“If "a" is the algebra generated by "C", then the sigma algebra generated by "a" and by "C" are the same.”
一看就是行家。
我理解了，如果用你说的 sigma-algebra contains algebra (both of them 阿are generated by the same "C"), 我就能理解你上面这句话了。因为可以证明
sigma-algebra generated by "C"=sc;
sigma-algebra generated by "a"=sa;
sc, sa 相互 contain. 对吧？
但是我没有理解，为什么 sigma-algebra generated by "C" is larger than the algebra generated by "C"?
按照定义，我怎么觉得 sigma-algebra 是一种带有那个可数集限制条件的，一种特殊的 algebra 呢？
问题1：
我觉得怎么 algebra 比 sigma-al
【在 a***s 的大作中提到】
:
: the algebra generated by "C", right? I'm not sure.
: No. The sigma-algebra generated by "C" is larger than the algebra generated
: by
: "C". You can say "a" is an algebra containing "C".
: algebra? I think "a" is unique since it is the "smallest". But then "C"
: generates the same algebra and sigma-algebra, right?
: It's the unique sigma-algebra.
: same algebra? is this generated algebra still "a"? does "a" and "C" generate
: the same sigma-algebra? is this generated sigma-algebra contained by "a"?
| a***s
发帖数: 616 | 4 Sigma algebra is much more complicate than algebra. So we usually start with
studying algebra and then extend the result to sigma algebra.
In constructing the Lebsgue measure, we want to measure as many sets as poss
ible and also make sure that intervals have measure in the common way, i.e.
[a,b) has measure b-a. (Let's focus on the whole set [0,1) to ,make things s
imple.)
So we start with interval of the form [a,b). Then we can easily extend to fi
nite union of intervals of this form, i.e. [a, | z****e
发帖数: 2024 | 5 太牛逼了。收藏了先。
您是教授吧？
最后，为什么 the sigma-algebra generated by "C", contains the algebra
generated by "C" 呢？
我怎么觉得凡是sigma-algebra就都是algebra，这样的话，不是反过来了吗？
谢谢。 | a***s
发帖数: 616 | 6 过奖了，别这样讲。
sigma algebra满足algebra的性质，但是algebra不一定有sigma algebra的性质。
比如通常讲的Borel sigma algebra当然是algebra: closed under complement, finit
e union, finite intersection。
但是[0,1)上由[a,b)型区间的finite union构成的collection只是一个algebra而不是s
igma algebra: not closed under countable union/intersection. e.g. intersecti
on over all n from 1 to infinity [0.5,0.5+1/n) gives {0.5} which is not in t
his collection.
【在 z****e 的大作中提到】
: 太牛逼了。收藏了先。
: 您是教授吧？
: 最后，为什么 the sigma-algebra generated by "C", contains the algebra
: generated by "C" 呢？
: 我怎么觉得凡是sigma-algebra就都是algebra，这样的话，不是反过来了吗？
: 谢谢。
| z****e
发帖数: 2024 | 7 瞻仰大侠风采。
本人才学分析，有些思维方式还没有建立。
多谢指点。
你这不当faculty就浪费了，虽然有不小难度。 |
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