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。Q: X=Y almost surely iff ∀A∈G∫AXdP=∫AYdP

the if direction:

https://math.stackexchange.com/questions/2494959/x-y-almost-surely-iff-forall-a-in-mathcalg-int-axdp-int-aydp?r=SearchResults&s=4|45.8271 

zhihu search:    math stack exchange

       math exchange

 

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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 GFG⊂F. Let X,YX,Y be GG-measurable, and integrable. Then, how does one prove that

X=YX=Y almost surely iff AGAXdP=AYdP∀A∈G∫AXdP=∫AYdP?

Here's my try: AXdP=AYdPAXYdP=A(XY)1[XY]+(YX)1[X<Y]dP=0∫AXdP=∫AYdP⇔∫AX−YdP=∫A(X−Y)1[X≥Y]+(Y−X)1[X

For A=[YX]A=[Y≥X], we get [XY]XYdP=0∫[X≥Y]X−YdP=0 which implies, by nonnegativity of XYX−Y on A, P(X=Y)=P(XY)P(X=Y)=P(X≥Y) or P(XY)=0P(X≥Y)=0

For A=[Y<X]A=[Y[X<Y]YXdP=0∫[XXYX−Y on A, P(X=Y)=P(X<Y)P(X=Y)=P(XP(X<Y)=0P(X

So, we get

(P(XY)=0  P(X=Y)=P(X<Y))  (P(X<Y)=0  P(X=Y)=P(XY))(P(X≥Y)=0 ∧ P(X=Y)=P(X

 

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]=[XY>0][X>Y]=[X−Y>0]

Is this a proper proof?

 

 

Who is BB, and what is meant by "Let X,YX,Y be random variables in GG?" That XX and YY are σ(G)σ(G)-measurable? – Math1000 Oct 30 '17 at 10:41
  •  
    @Math1000 I've corrected the typos. – An old man in the sea. Oct 30 '17 at 11:49

 

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