Is the Vitali set measurable?

A Vitali set can not be included in the family of measurable sets for any locally finite translation invariant measure (except for the zero measure). In particular it is not Lebesgue measurable.

Is the Vitali set Borel?

Topological proof that a Vitali set is not Borel.

Is the Vitali set uncountable?

This cardinal number is definitely uncountable: 2c>c>ℵ0. so by Cantor-Bernstein Theorem all cardinal numbers in this chain of inequalities are the same.

What is the outer measure of the Vitali set?

Notice that the outer measure of a set is always defined. What is the outer measure of the Vitali set V we constructed? It cannot be 0 or 1, but has to be between 0 and 1.

How do you show a set is not measurable?

Let Q be the set of all rational numbers. Then a set X( called a Vitali set) having in accordance with the axiom of choice exactly one element in common with every set of the form Q+a, where a is any real number, is non-measurable.

Is the set of reals measurable?

Any closed interval [a, b] of real numbers is Lebesgue-measurable, and its Lebesgue measure is the length b − a. The Cantor set and the set of Liouville numbers are examples of uncountable sets that have Lebesgue measure 0. If the axiom of determinacy holds then all sets of reals are Lebesgue-measurable.

Are the reals measurable?

In particular, the Lebesgue measure of the set of algebraic numbers is 0, even though the set is dense in R. The Cantor set and the set of Liouville numbers are examples of uncountable sets that have Lebesgue measure 0. If the axiom of determinacy holds then all sets of reals are Lebesgue-measurable.

Are the rationals measurable?

Since the intervals overlap , the measure of the Rationals is bounded by the unions of these intervals with total measure zero. Then, by completeness of the measure a subset of measure zero (Rationals) are measurable with measure zero.

Which phenomenon is not measurable?

Consistent definitions of measure and probability Some sets might be tagged “non-measurable”, and one would need to check whether a set is “measurable” before talking about its volume. The axioms of ZFC (Zermelo–Fraenkel set theory with the axiom of choice) might have to be altered.

Are open sets measurable?

Since the outer measure of an interval is its length, and intervals are now measurable, their (Lebesgue) measure must also be their length. Therefore open sets are measurable. But closed sets are the complements of open sets, and complements of measurable sets are measurable. Therefore closed sets are measurable.