Proof of nonmeasurable set

• cerenacerenb
In summary, the conversation discusses a proof involving sets with finite outer measure that are not measurable. The inner measure of a set is non-negative and less than the outer measure, and if a set is not measurable, the inner measure must be less than the outer measure. The idea of using an open set, containing the set in question, that is measurable is also mentioned.

cerenacerenb

1. Let a set E have a finite outer measure. Show that if E is not measurable, then there is an open set O containing the set that has finite outer measure and for which m*(O~E) > m*(O)-m*(E).

Any idea how to takle this proof?

You waited two whole minutes for a response before bumping?

A set is measurable if and only if both outer and inner measure are finite and equal. The inner measure is non-negative and less than the outer measure so since the outer measure is finite, so is the inner measure. That means that, since this set is not measurable, the inner measure must be less than the outer measure. What if O is an open set, containing E, that is measurable?

What is a proof of nonmeasurable set?

A proof of nonmeasurable set is a mathematical proof that demonstrates the existence of a set that cannot be assigned a measure. In other words, there is no way to assign a numerical value to the size or volume of this set.

Why is the concept of a nonmeasurable set important?

The existence of nonmeasurable sets is an important concept in measure theory, which is a branch of mathematics that deals with assigning numerical values to the size or volume of sets. The existence of nonmeasurable sets challenges the traditional understanding of measure, and has implications for other areas of mathematics such as probability theory and analysis.

How is a proof of nonmeasurable set constructed?

A proof of nonmeasurable set is typically constructed using a technique known as the Vitali construction. This involves dividing a larger set into smaller, disjoint subsets and then using the Axiom of Choice to choose one element from each subset. This process results in a nonmeasurable set.

Can you give an example of a nonmeasurable set?

One example of a nonmeasurable set is the Vitali set, which was first constructed by Giuseppe Vitali in 1905. This set consists of all possible combinations of rational numbers in the interval [0,1], where two numbers are considered equivalent if their difference is a rational number. This set is nonmeasurable because it cannot be assigned a numerical measure due to its infinite and disjoint nature.

What are the implications of the existence of nonmeasurable sets?

The existence of nonmeasurable sets has implications for other areas of mathematics, such as probability theory and analysis. It also challenges our traditional understanding of measure and highlights the importance of the Axiom of Choice in constructing these sets. Additionally, the existence of nonmeasurable sets has philosophical implications for the concept of infinity and our understanding of the nature of mathematical objects.