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Darío G
Darío G
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Let $(X,\mu)$ be a probability space, and $0<\epsilon<1/2$. Let $\{A_i:i\in /\mathbb{N}\}$$\{A_i:i\in \mathbb{N}\}$ be a collection of measurable subsets of $X$ such that $\mu(A_i)\geq \epsilon$ for all $i\in\mathbb{N}$.

Is it always true that there are indices $i<j$ such that $\mu(A_i\cap A_j)\geq \epsilon^2$ ? Is it possible to classify the (possible) counterexamples?

Let $(X,\mu)$ be a probability space, and $0<\epsilon<1/2$. Let $\{A_i:i\in /\mathbb{N}\}$ be a collection of measurable subsets of $X$ such that $\mu(A_i)\geq \epsilon$ for all $i\in\mathbb{N}$.

Is it always true that there are indices $i<j$ such that $\mu(A_i\cap A_j)\geq \epsilon^2$ ? Is it possible to classify the (possible) counterexamples?

Let $(X,\mu)$ be a probability space, and $0<\epsilon<1/2$. Let $\{A_i:i\in \mathbb{N}\}$ be a collection of measurable subsets of $X$ such that $\mu(A_i)\geq \epsilon$ for all $i\in\mathbb{N}$.

Is it always true that there are indices $i<j$ such that $\mu(A_i\cap A_j)\geq \epsilon^2$ ? Is it possible to classify the (possible) counterexamples?

Source Link
Darío G
Darío G
  • 167
  • 9

Measure of intersections in probability spaces

Let $(X,\mu)$ be a probability space, and $0<\epsilon<1/2$. Let $\{A_i:i\in /\mathbb{N}\}$ be a collection of measurable subsets of $X$ such that $\mu(A_i)\geq \epsilon$ for all $i\in\mathbb{N}$.

Is it always true that there are indices $i<j$ such that $\mu(A_i\cap A_j)\geq \epsilon^2$ ? Is it possible to classify the (possible) counterexamples?