Prove that no finite measure space can have uncountably many members in its $\sigma$-algebra with strictly positive measure.

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In summary, a measure space is a set with a measure function that assigns values to subsets, and it has uncountably many members if it contains an infinite number of elements that cannot be counted. A sigma-algebra is a collection of subsets that satisfies certain properties and is used in measure theory to define measurable sets. It is impossible for a finite measure space to have uncountably many members in its sigma-algebra, and a measure is said to have strictly positive measure if every non-empty subset has a positive measure. This can be proven by contradiction.
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Here is this week's POTW:

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Prove that no finite measure space can have uncountably many members in its $\sigma$-algebra with strictly positive measure.

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No one answered this week’s problem. You can read my solution below.

Suppose $(X,M,\mu)$ is a finite measure space with uncountably many $A\in M$ such that $\mu(A) > 0$. There exists an $n \in \Bbb N$ such that uncountably many $A\in M$ with $\mu(A) > 1/n$. If $A$ is the countable disjoint union of sets $A_i\in M$ of measure greater than $1/n$, then $A\in M$ and $\mu(A) = \sum \mu(A_i) = \infty$. This is a contradiction.
 

Related to Prove that no finite measure space can have uncountably many members in its $\sigma$-algebra with strictly positive measure.

What is a finite measure space?

A finite measure space is a mathematical concept used in measure theory to describe a set with a finite measure, or size. It consists of a set of elements and a measure function that assigns a numerical value to each subset of the set.

What is a $\sigma$-algebra?

A $\sigma$-algebra is a collection of subsets of a set that satisfies certain properties. In the context of measure theory, it is used to define which subsets of a set are measurable and can have a measure assigned to them.

Why can't a finite measure space have uncountably many members in its $\sigma$-algebra with strictly positive measure?

This is because a finite measure space, by definition, has a finite measure. This means that the sum of the measures of all the subsets in the $\sigma$-algebra must be finite. If there were uncountably many subsets with strictly positive measure, the sum would be infinite, which contradicts the finite measure of the space.

Can a finite measure space have countably infinite members in its $\sigma$-algebra with strictly positive measure?

No, a finite measure space cannot have countably infinite members in its $\sigma$-algebra with strictly positive measure. This is because the sum of the measures of countably infinite subsets with strictly positive measure would still be infinite, which contradicts the finite measure of the space.

What is the significance of this statement in mathematics?

This statement is significant in mathematics because it highlights the limitations of finite measure spaces and their $\sigma$-algebras. It also demonstrates the importance of understanding the properties and definitions of mathematical concepts, such as measure spaces and $\sigma$-algebras, in order to make valid mathematical arguments and proofs.

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