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Working on probabilistic algorithms with real valued inputs and I stumbled upon a question that I cannot answer. I ask here, in the hope that my inexperience with measure theory will not make anybody frown :)

I have two probability measures $P,P'$ defined over the same $\sigma$-algebra $\Sigma$ that is generated by a $\pi$-system $G$. I know that $|P(g)-P'(g)| \le e$, for each $g \in G$. Can I conclude that there is a bound (any bound) on $|P(s) - P'(s)|$ for every set $s \in \Sigma$? If no, is there any additional property that would guarantee the existence of the bound?

Thank to those of you that will answer.

Grey Wolf
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  • I don't think you can say much in general. Consider the case where $\pi$ consists precisely of all sets that have at most one element. – PhoemueX May 15 '22 at 15:02
  • Thanks :). Would it make a difference if the generators form an algebra? – Grey Wolf May 15 '22 at 15:14

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If $G$ is only a $\pi$-system, the theorem is not true. Counterexamples can be found when the probability space $\Omega$ is a finite set and $G = \{A \subset \Omega : |A| \leq 1\} \cup \{\Omega\}$.

If $G$ is an algebra, then the theorem is true, i.e. $|P(A) - P'(A)| \leq e$ for all $A \in G$ implies $|P(A) - P'(A)| \leq e$ for all $A \in \Sigma$.

Proof: Let $A \in \Sigma$ be arbitrary. Let $h > 0$ be arbitrary. Since $G$ is an algebra and $P + P'$ is a finite measure, there is $A_h \in G$ such that $\|1_{A_h} - 1_{A}\|_{L^1(P + P')} \leq h$ (proof sketch: use the $\pi$-$\lambda$ theorem). So \begin{align} |P(A) - P'(A)| &\leq |P(A) - P(A_h)| + |P(A_h) - P'(A_h)| + |P'(A_h) - P'(A)| \\ &\leq h + e + h. \end{align} Letting $h \searrow 0$ yields $|P(A) - P'(A)| \leq e$.

Mason
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  • Thank you very much :). I am indeed dealing with an algebra of sets.

    This property is kind of cool anyway. It shows how to get define approximate measures starting from the generators.

    Thank you again :)

     – Grey Wolf May 16 '22 at 11:55