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YCor
YCor
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Union Ordered union of ordered Borel sets

Let $\mathfrak{A}$ be an uncountable collection of Borel sets in $\mathbb{R}^d$ such that for any $A,B\in\mathfrak{A}$, either $A\subset B$ or $A\supset B$. Then is it necessarily true that the union $\cup_{A\in\mathfrak{A}}A$$\bigcup_{A\in\mathfrak{A}}A$ is Borel measurable?

Union of ordered Borel sets

Let $\mathfrak{A}$ be an uncountable collection of Borel sets in $\mathbb{R}^d$ such that for any $A,B\in\mathfrak{A}$, either $A\subset B$ or $A\supset B$. Then is it necessarily true that the union $\cup_{A\in\mathfrak{A}}A$ is Borel measurable?

Ordered union of Borel sets

Let $\mathfrak{A}$ be an uncountable collection of Borel sets in $\mathbb{R}^d$ such that for any $A,B\in\mathfrak{A}$, either $A\subset B$ or $A\supset B$. Then is it necessarily true that the union $\bigcup_{A\in\mathfrak{A}}A$ is Borel measurable?

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user137602
user137602
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union Union of ordered Borel sets

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user137602
user137602
  • 53
  • 3

union of ordered Borel sets

Let $\mathfrak{A}$ be an uncountable collection of Borel sets in $\mathbb{R}^d$ such that for any $A,B\in\mathfrak{A}$, either $A\subset B$ or $A\supset B$. Then is it necessarily true that the union $\cup_{A\in\mathfrak{A}}A$ is Borel measurable?