MHB Probability measure: prove or disprove

AI Thread Summary
The discussion revolves around proving or disproving several properties of a probability measure \( Q \) over a \( \sigma \)-algebra \( \mathcal{B} \). Initial attempts to validate the statements reveal contradictions, particularly when testing with specific sets like \( A = \emptyset \) and \( B = \Omega \), leading to the conclusion that some statements do not hold in general. The participants explore the implications of the properties, confirming that \( Q(\emptyset) = 0 \) is valid under the definition of a probability measure. The conversation emphasizes the importance of disjoint sets in the context of probability measures. Overall, the exploration highlights the complexities and nuances in the properties of probability measures.
mathmari
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Hey! :o

I want to prove or disprove that for a probability measure $Q$ over a $\sigma$-algebra $\mathcal{B}$ with $A,B\in \mathcal{B}$ the following hold:
  1. $Q(A\cup B)=1-Q(\overline{A}\cup \overline{B})$
  2. $1-Q(\overline{A}\cap \overline{B})=Q(\overline{A})+Q(\overline{B})+Q(\overline{A}\cup \overline{B})$
  3. $1-Q(\overline{A})-Q(\overline{B})=Q(A)-1+Q(B)$
  4. $Q(A)\neq \emptyset\Rightarrow A\neq \emptyset$
I have done the following :

  1. Let $\Omega$ be the universal set.
    We suppose that the statement is true.
    For $A=\emptyset$ and $B=\Omega$ be we get the following:
    \begin{align*}&Q(\emptyset\cup \Omega)=1-Q(\overline{\emptyset}\cup \overline{\Omega}) \\ & \Rightarrow Q(\Omega)=1-Q(\Omega\cup \emptyset) \\ & \Rightarrow Q(\Omega)=1-Q(\Omega) \\ & \Rightarrow 2Q(\Omega)=1 \\ & \Rightarrow Q(\Omega)=\frac{1}{2} \\ & \Rightarrow 1=\frac{1}{2}\end{align*}
    So, the statement is in general not true.

    Is everything correct?
    $$$$
  2. Let $S\in \mathcal{B}$.
    We have that $\Omega = S\cup \overline{S}$. Then $Q(\Omega)=Q(S\cup \overline{S})$. Since $Q$ is a probability measure, we get that $Q(\Omega)=1$ and $Q(S\cup \overline{S})=Q(S)+Q(\overline{S})$.
    Therefore, we get that:
    $$Q(\Omega)=Q(S\cup \overline{S}) \Rightarrow 1=Q(S)+Q(\overline{S}) \Rightarrow Q(\overline{S})=1-Q(S)$$

    By De Morgan’s laws we have that $\overline{A\cup B}=\overline{A}\cap \overline{B}$.

    We have that $Q(\overline{A\cup B})=1-Q(A\cup B) \Rightarrow Q(\overline{A}\cap \overline{B})=1-Q(A\cup B) \Rightarrow 1-Q(\overline{A}\cap \overline{B})=Q(A\cup B)$.

    So, we have to check if $Q(A\cup B)=Q(\overline{A})+Q(\overline{B})+Q(\overline{A}\cup \overline{B})$, right? How could we check that?

    $$$$
  3. We have that $Q(\overline{A})=1-Q(A)$ and $Q(\overline{B})=1-Q(B)$.

    So, we get:
    $$1-Q(\overline{A})-Q(\overline{B})=1-(1-Q(A))-(1-Q(B))=1-1+Q(A)-1+Q(B)=Q(A)-1+Q(B)$$
    So, the statement is true.

    Is everything correct?

    $$$$
  4. Could you give me a hint for this statement?
 
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mathmari said:
Hey! :o

I want to prove or disprove that for a probability measure $Q$ over a $\sigma$-algebra $\mathcal{B}$ with $A,B\in \mathcal{B}$ the following hold:
  1. $Q(A\cup B)=1-Q(\overline{A}\cup \overline{B})$
  2. $1-Q(\overline{A}\cap \overline{B})=Q(\overline{A})+Q(\overline{B})+Q(\overline{A}\cup \overline{B})$
  3. $1-Q(\overline{A})-Q(\overline{B})=Q(A)-1+Q(B)$
  4. $Q(A)\neq \emptyset\Rightarrow A\neq \emptyset$

Hey mathmari!

1. All correct. (Nod)

2. How about checking for, say, $A=B=\Omega$?

3. Good.

4. The statement $S\Rightarrow T$ is true if and only if $\lnot T \Rightarrow \lnot S$.
Can we prove the latter? (Wondering)
 
I like Serena said:
2. How about checking for, say, $A=B=\Omega$?

For $A=B=\Omega$ we get the following:

The left side is equal to $1-Q(\overline{\Omega}\cap \overline{\Omega})=1-Q(\emptyset\cap \emptyset)=1-Q(\emptyset)=1-0=1$.

The right side is equal to $Q(\overline{\Omega})+Q(\overline{\Omega})+Q(\overline{\Omega}\cup \overline{\Omega}) =Q(\emptyset)+Q(\emptyset)+Q(\emptyset\cup \emptyset)=0+0+Q(\emptyset)=0$.

Therefore, the statement does not hold in general, right? (Wondering)
I like Serena said:
4. The statement $S\Rightarrow T$ is true if and only if $\lnot T \Rightarrow \lnot S$.
Can we prove the latter? (Wondering)

So, we have to prove that $\lnot A\neq \emptyset \Rightarrow \lnot Q(A)\neq 0$, i.e. that $A= \emptyset \Rightarrow Q(A)=0$, i.e. $Q(\emptyset )=0$, right? (Wondering)

Since $Q$ is a probability measure, we have that $Q(\Omega)=1$ and that $Q(S_1\cup S_2)=Q(S_1)+Q(S_2)$, for $S_1, S_2\in \mathcal{B}$.
So, we get the following:
$$Q(\Omega)=Q(\Omega\cup \emptyset)=Q(\Omega)+Q(\emptyset) \Rightarrow 1=1+Q(\emptyset) \Rightarrow Q(\emptyset)=0$$
right?

So, since it holds that $\lnot A\neq \emptyset \Rightarrow \lnot Q(A)\neq 0$, it follows that $Q(A)\neq 0\Rightarrow A\neq \emptyset$, right? (Wondering)
 
mathmari said:
Since $Q$ is a probability measure, we have that $Q(\Omega)=1$ and that $Q(S_1\cup S_2)=Q(S_1)+Q(S_2)$, for $S_1, S_2\in \mathcal{B}$.

We have the additional constraint that $S_1$ and $S_2$ must be disjoint. (Nerd)

Otherwise everything is correct. (Mmm)
 
I like Serena said:
We have the additional constraint that $S_1$ and $S_2$ must be disjoint. (Nerd)

Otherwise everything is correct. (Mmm)

Great! Thank you! (Yes)
 
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