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几乎处处.
在测度论[注 1]里,若说一个性质为几乎处处成立,即表示不符合此性质的元素组成的集合为一零测集,即其测度等于零的集合。当使用在实数的性质上时,若没有另外提起则假定为勒贝格测度。[注 2]
一个有全测度的集合是一个其补集为零测度的集合。
除了说一个性质几乎处处成立之外,偶尔亦可以说一个性质是对几乎所有元素成立的,即使几乎所有这一词有着其他的意义。
下面是包含有“几乎处处”这一词的一些定理:
- 若
:
→
为一勒贝格可积函数且
几乎处处大于零,则
。
- 若
:
→
为一单调函数,则
几乎处处可微。
- 当
:
→
为勒贝格可积且对所有实数
,
![{\displaystyle \int _{a}^{b}|f(x)|\,dx<\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e35f79e8c4bce06421ca9b54524c5024f765801d)
- 则存在一零集E(根据
)使得若
不在
内,其勒贝格平均
![{\displaystyle {\frac {1}{2\epsilon ))\int _{x-\epsilon }^{x+\epsilon }f(t)\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5aaa4bc8b2b4a8639d52be334734301a3505e54)
- 便会收敛至
,当ε趋向至零时。换句话说,
的勒贝格平均几乎处处收敛至
。集合E则称为
的勒贝格集合,且可以证明为零测度的。
- 若
在
上为博雷尔可测的,则对几乎所有
,函数
→
为博雷尔可测的。
- 一有界函数
:
->
为黎曼可积的,当且仅当其为几乎处处连续的。
在实分析之外,“几乎处处”一词可以用极大滤子定义。例如在超实数的建构中,一个超实数被定义为相对于某一滤子几乎处处相等的等价类。
在抽象代数及其相关领域中,“几乎处处”通常指某性质只对给定集合中的有限个元素不成立。
在概率论里,这一词变成了“几乎必然”,“几乎确定”或“几乎总是”,相对于一为1的概率。
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