测度与积分 Measures and Integration学习笔记

希望能坚持学习下去

可测空间 Measurable Spaces

$\sigma$-algebra

algebra 和 $\sigma$-algebra 区别首先看两个的定义：

1. An algebra is a collection of subsets closed under finite unions and intersections.
2. A sigma algebra is a collection closed under countable unions and intersections.

唯一的区别在于finite 和 countable。
一般来讲，finite unions 就是collection 中有限个element (这里的元素是指集合)的并（union），数学符号可以表示为 ${\bigcap }_{i=1}^k{A}_i$, 其中 $k$ 为正整数。而countable unions 表示为 ${\bigcap }_{i=1}^{\infty }{A}_i$。countable 包含可数的无限，有无穷多的整数， There are infinitely many integers, 所以countable 是比finite 更多的。某种意义上可以理解为 最小的infinity。

上述的情况通常在开集和闭集中涉及到。A finite union of closed sets is closed. 但是 An inifinite union of closed sets可能不是closed.
一个简单的例子就是：collection of sets {$I_n=[\frac{1}{n},1-\frac{1}{n}]$}, 可以看到每个set $I_n$都是closed. 但是如果考虑${\bigcap }_{i\in \mathbb{N}}{I}_i=(0,1)$, 显然是 open set.

algebra 和 $\sigma$-algebra 都是集合的collections. To be closed under finite intersections means that taking any number of finite intersections of elements of the algebra yields an element (another set) that is in the algebra. But maybe this isn’t true for an infinite intersection, etc.

这部分参考链接: Sigma algebra and algebra difference