Measure zero

Definition

A set is considered to have measure zero if, intuitively speaking, it is so small that it can be covered by a collection of intervals or sets whose total length can be made arbitrarily small. This concept is essential in understanding the properties of critical values and the application of Sard's theorem, which relates to the behavior of smooth functions and their critical points. Measure zero sets play a significant role in analysis and topology, particularly when examining properties of functions and differentiable maps.

5 Must Know Facts For Your Next Test

1. Measure zero sets can be uncountably infinite, yet they have 'small' size in terms of measure theory; for example, the Cantor set has measure zero despite being uncountable.
2. Sard's theorem states that the set of critical values (where derivatives vanish) of a smooth function has measure zero, meaning most values are not critical.
3. The concept of measure zero is essential in real analysis as it helps identify sets that do not affect integrals and other measures significantly.
4. Sets with measure zero can be ignored when considering the almost everywhere properties of functions, allowing for exceptions in integration and differentiability.
5. In many cases, properties such as continuity and differentiability can be analyzed by focusing on sets with full measure instead of those with measure zero.

Review Questions

• How does the concept of measure zero relate to Sard's theorem and the critical values of smooth functions?
  • Sard's theorem asserts that the critical values of a smooth function form a set with measure zero. This means that while there may be critical points where the derivative is zero or undefined, these points are rare in the sense that they occupy no 'volume' in the range of the function. Consequently, most values that a smooth function can take are non-critical, highlighting how critical points do not significantly affect the overall behavior of the function.
• Discuss why sets with measure zero are important when evaluating properties of differentiable functions.
  • Sets with measure zero are crucial in evaluating properties such as continuity and integrability because they allow mathematicians to focus on almost everywhere behavior without being hindered by 'small' exceptions. For instance, if a function is continuous almost everywhere except on a set of measure zero, it still retains significant properties in most parts. This understanding simplifies analysis and helps establish broader conclusions about functions without needing to account for negligible sets.
• Evaluate how the concept of measure zero impacts our understanding of critical points and their influence on smooth mappings.
  • The notion of measure zero fundamentally alters our perspective on critical points within smooth mappings by indicating that these points are not only rare but also not significant in terms of their contribution to the overall mapping behavior. By establishing that the set of critical values has measure zero, we conclude that smooth functions behave 'normally' or predictably most of the time. This distinction allows for deeper insights into topology and differential geometry, where understanding what occurs away from critical points becomes essential for grasping the structure and characteristics of mappings.

Related terms

Lebesgue Measure: A method of assigning a measure (or size) to subsets of n-dimensional space, generalizing the concept of length, area, and volume.

Borel Set: A set that can be formed from open or closed intervals through countable unions, intersections, and complements, which are important in measure theory.

Critical Point:

A point in the domain of a function where the derivative is either zero or undefined, often leading to local maxima or minima.