E{XE{Y|C}}=E{YE{X|C}}
现在有没有适合大学生用的搜题软件呢? https://www.zhihu.com/question/51935291/answer/514312093 Approach0 (https://approach0.xyz/search/) 专门用来搜索 Math StackExchange (支持数学公式)。
the if direction:
zhihu search: math stack exchange
math exchange
2,slader: 这个软件是帮助你写作业的,一般的教科书后面会有答案,但不会有详解,这个软件会帮助你更好的学习。
手机上当然首推Desmos,Wolfram Alpha还有Brilliant啦!
Math Lab。能解微分方程能画心型图神器
设x,y是概率空间(Ω,F,P)上的拟可积随机变量,证明:X=Y a.e 当且仅当 xdp = ydp 对每个A∈F成立
Let (Ω,F,P)(Ω,F,P) be a probability space with G⊂FG⊂F. Let X,YX,Y be GG-measurable, and integrable. Then, how does one prove that
X=YX=Y almost surely iff ∀A∈G∫AXdP=∫AYdP∀A∈G∫AXdP=∫AYdP?
Here's my try: ∫AXdP=∫AYdP⇔∫AX−YdP=∫A(X−Y)1[X≥Y]+(Y−X)1[X<Y]dP=0∫AXdP=∫AYdP⇔∫AX−YdP=∫A(X−Y)1[X≥Y]+(Y−X)1[X<Y]dP=0
For A=[Y≥X]A=[Y≥X], we get ∫[X≥Y]X−YdP=0∫[X≥Y]X−YdP=0 which implies, by nonnegativity of X−YX−Y on A, P(X=Y)=P(X≥Y)P(X=Y)=P(X≥Y) or P(X≥Y)=0P(X≥Y)=0
For A=[Y<X]A=[Y<X], we get ∫[X<Y]Y−XdP=0∫[X<Y]Y−XdP=0 which implies, by nonnegativity of X−YX−Y on A, P(X=Y)=P(X<Y)P(X=Y)=P(X<Y) or P(X<Y)=0P(X<Y)=0
So, we get
which gives P(X=Y)=1P(X=Y)=1.
I can choose A as above since the sum of measurable functions is also measurable [X>Y]=[X−Y>0][X>Y]=[X−Y>0]
Is this a proper proof?
- @Math1000 I've corrected the typos. – An old man in the sea. Oct 30 '17 at 11:49