Question

Let \(A \subset[0,1]\) be a Borel set such that \(0

Short answer

Functions F and G display unique behavior due to set properties: F is increasing with F' = 0 on a positive measure set, and G is continuous but non-monotonic.

Step-by-step solution

Understand Context

We are given a set \(A\) which is a subset of \([0,1]\) and satisfies the property \(0 < m(A \cap I) < m(I)\) for every subinterval \(I\) of \([0,1]\). Here, \(m\) indicates Lebesgue measure, making this a measure-theoretic problem involving a measure function on Borel sets.

Introduce Function F

Consider the function \(F(x) = m([0, x] \cap A)\), which represents the measure of the intersection of \([0, x]\) with \(A\). As half-open intervals and Borel sets are involved, we define this using Lebesgue measure.

Properties of F

We need to prove that \(F\) is absolutely continuous and strictly increasing. Since \(0 < m(A \cap I)\) for any subinterval, the measure increases over any interval. \(F\) is absolutely continuous as it represents a Lebesgue integral over \(A\), a Borel set.

F Prime is Zero on Positive Measure Set

Although \(F\) is strictly increasing, \(F' = 0\) on a set of positive measure. This paradoxical condition can happen because \(F\) is the distribution function of a singular measure, hence it changes without the derivative ever differing from zero on a set with positive measure.

Introduce Function G

Define \(G(x) = m([0, x] \cap A) - m([0, x] \setminus A)\). This function's absolute continuity follows from the properties of measures and linearity of subtraction, but analysis of monotonicity is required.

Non-Monotonicity of G

Show that \(G(x)\) cannot be monotonic on any subinterval \(I \subset [0,1]\). Since both terms \(m([0,x] \cap A)\) and \( -m([0,x] \setminus A)\) vary independently as \(A\) was defined with \(0 < m(A \cap I) < m(I)\), this independence prevents overall monotonic behavior of \(G\).

Key concepts

Lebesgue measure

The Lebesgue measure is an essential concept in modern mathematics, particularly in measure theory and probability. It extends the idea of length, area, and volume to more complex sets, beyond standard geometric shapes. It's vital for defining the size of a set in the real number line, including irregularly shaped sets.

In the context of our exercise, Lebesgue measure, denoted usually by \(m\), allows us to assign a measure to Borel sets such as our set \(A \subset [0,1]\). This measure provides a way to "weigh" subsets of the interval \([0,1]\) and evaluate properties like continuity and differentiability of functions defined by these measures.

Absolutely continuous functions, like \(F(x)\) and \(G(x)\) in the exercise, rely on Lebesgue measure. A function is said to be absolutely continuous if there exists an epsilon for every delta such that the difference in measures over corresponding \(x\)-values is arbitrarily small, ensuring smoothness in the function's behavior.

• It captures distribution and concentration of values within specific ranges.
• This type of function enables us to consider integrals and derivatives in a more nuanced way than traditional calculus allows, highlighting the adaptability of Lebesgue measure.

Borel sets

Borel sets form the backbone of measure theory, as they include all possible open and closed sets that can be constructed on the real number line, including any operations such as countable unions or intersections. Understanding Borel sets is crucial because they are the simplest measurable sets under the Lebesgue measure.

In our exercises, set \(A\) is defined as a Borel set, meaning it's one of these well-defined sets within \([0,1]\), capable of being measured using Lebesgue measure. This set is distinguished because its intersection with any subinterval \(I\) satisfies the condition \(0 < m(A \cap I) < m(I)\).

This condition implies that the size of the intersection is non-zero but not equal to the whole interval \(I\), representing a non-trivial example of a Borel set.

• They are pivotal to understanding functions like \(F\) and \(G\) in the exercise because these functions are defined using intersections involving Borel sets.
• Working with Borel sets and the Lebesgue measure allows us a comprehensive approach to measure theory, enabling us to tackle complex problems like differentiating and integrating across exotic subsets of \([0,1]\).

singular measure

A singular measure represents an intriguing scenario in measure theory. It describes a measure that is concentrated on a set of Lebesgue measure zero, creating situations where a function can increase without a corresponding change in its derivative. This concept is crucial to grasp for functions that change abruptly but still maintain zero derivative almost everywhere.

In the problem, the function \(F(x)\) accounts for being absolutely continuous and increasing, yet we find that its derivative \(F' = 0\) on a set of positive measure, which is unexpected. This fits the profile of a singular measure—while the intuitive understanding of a derivative suggests zero change, a singular measure allows for value changes over "invisible" subsets.

Such measures also explain why functions, though continuous and smooth over intervals, can break traditional calculus rules.

• They emphasize the need to go beyond typical differentiations, showcasing the diverse behavior of functions under more generalized measures.
• By exploring singular measures, we gain deeper insight into phenomena such as continuous functions without a standard derivative, impacting how we interpret measurable and integrable functions.