Measure Zero

Definition

A set is said to have measure zero if, intuitively, it occupies no space in the real number line, meaning it can be covered by a countable collection of intervals whose total length can be made arbitrarily small. Measure zero sets are significant because they provide insight into the properties of Riemann integrable functions, particularly in understanding when a function may fail to be integrable due to the presence of such sets. They play a crucial role in the discussion of functions that are Riemann integrable, as functions that differ from Riemann integrable functions only on measure zero sets can still be integrated without affecting the integral's value.

5 Must Know Facts For Your Next Test

1. Measure zero sets can be finite or countably infinite, meaning they can consist of isolated points or even countable unions of such points.
2. The property of having measure zero is preserved under countable unions and intersections, which means if you take any countable collection of measure zero sets, their union is also measure zero.
3. In terms of Riemann integration, a bounded function that is discontinuous only on a set of measure zero is still Riemann integrable.
4. Examples of measure zero sets include any finite set, countable subsets of the real line like the rational numbers, and Cantor sets constructed with certain properties.
5. The existence of measure zero sets allows for a richer structure in real analysis and helps in proving key results such as the Lebesgue Dominated Convergence Theorem.

Review Questions

• How does the concept of measure zero relate to the Riemann integrability of functions?
  • Measure zero sets are crucial in determining Riemann integrability because if a function is discontinuous only on a set of measure zero, it can still be classified as Riemann integrable. This means that such discontinuities do not impact the overall area under the curve represented by the integral. Therefore, understanding measure zero helps in identifying when functions maintain their integrability despite certain exceptions.
• Discuss the implications of having multiple measure zero sets when analyzing Riemann integrable functions.
  • When dealing with multiple measure zero sets in the context of Riemann integrable functions, it’s important to recognize that their union will also have measure zero. This implies that if a function has discontinuities across several measure zero sets, we can still consider its Riemann integrability intact. Consequently, we can conclude that discontinuities spread across these sets do not contribute to altering the value of an integral, highlighting how flexibility in function behavior exists within these boundaries.
• Evaluate how understanding measure zero contributes to broader theories in mathematical analysis and integration techniques.
  • Understanding measure zero greatly enhances broader theories in mathematical analysis by bridging gaps between different types of integration techniques. It lays groundwork for concepts like Lebesgue integration, where the focus shifts from pointwise behavior to measure-theoretic properties. The ability to navigate between Riemann and Lebesgue integrals by recognizing sets that do not impact integral values allows mathematicians to tackle more complex problems with confidence. This deepens our grasp on convergence, continuity, and functional behaviors across different spaces.

Related terms

Lebesgue Measure: A systematic way of assigning a measure to subsets of n-dimensional space, extending the concept of length, area, and volume.

Riemann Integral:

A method of assigning a number to a function's integral over an interval, using limits of sums of function values at specified points multiplied by small interval widths.

Borel Set: Any set that can be formed from open or closed intervals through countable unions, intersections, and relative complements.