Cardinality vs. Measure

The concepts of Cardinality and Measure are fundamental in mathematics, offering two different perspectives on the “size” of a set. These concepts often coincide in our intuition, especially when dealing with finite sets, but diverge significantly in the context of infinite sets, revealing deep and non-intuitive properties of mathematical structures. Understanding the distinction between cardinality and measure is crucial for navigating areas like set theory, measure theory, and real analysis.

Cardinality

Cardinality refers to the number of elements in a set, serving as a measure of the set’s size in terms of its elements. The concept is straightforward for finite sets, but for infinite sets, cardinality is defined in terms of one-to-one correspondences (bijections) between sets.

• Finite Sets: The cardinality is simply the count of elements. For example, the set $A=\{1,2,3\}$ has a cardinality of 3.
• Infinite Sets: Two sets are said to have the same cardinality if there exists a bijection between them. For instance, the set of natural numbers $\mathbb{N}$ and the set of integers $\mathbb{Z}$ have the same cardinality, despite $\mathbb{Z}$ seemingly being “larger” because a bijection can be established between them.

Levels of Infinity: Cantor’s theorem shows that not all infinite sets have the same cardinality. For example, the set of real numbers $\mathbb{R}$ has a strictly greater cardinality than the set of natural numbers $\mathbb{N}$, indicating different “sizes” of infinity.

Measure

Measure quantifies the “size” of a set in terms of length, area, volume, or other continuous quantities, extending beyond the mere count of elements. In the context of real numbers, the Lebesgue measure is commonly used, assigning a length to subsets of $\mathbb{R}$.

• Lebesgue Measure: This is a way to assign a non-negative real number to (ideally) any subset of ${\mathbb{R}}^n$ in a way that generalizes the concept of length, area, and volume. The measure of an interval $(a,b)$ in $\mathbb{R}$, for example, is defined to be its length $b-a$.
• Properties: A measure must satisfy certain properties, such as being countably additive. If $\{A_i{\}}_{i=1}^{\infty }$ is a countable collection of disjoint measurable sets, then the measure of their union is equal to the sum of their measures: $m\left({\bigcup }_{i=1}^{\infty }{A}_i\right)={\sum }_{i=1}^{\infty }m(A_i)$.

Cardinality vs. Measure: Key Differences

1. Nature: Cardinality is about counting discrete elements, while measure deals with continuous sizes.
2. Infinite Sets: Cardinality can show equivalence between sets that seem intuitively different in size (e.g., $\mathbb{N}$ and $\mathbb{Z}$), while measure can assign a “size” to subsets of $\mathbb{R}$, such as intervals or more complex constructions like the Cantor set.
3. Non-intuitive Results:
   • The Cantor set has the same cardinality as $\mathbb{R}$ (both are uncountably infinite) but has a Lebesgue measure of 0, illustrating how cardinality and measure capture fundamentally different notions of “size”.
   • There exist sets (like the Vitali set) that are not Lebesgue measurable, showcasing the limitations of measure theory in capturing the “size” of certain sets.

Conclusion

While both cardinality and measure offer insights into the concept of size within mathematical sets, they do so from different angles and with different implications, especially in the context of infinite sets. The distinction between these concepts is crucial in understanding the structure of Mathematical Spaces, the behavior of functions across these spaces, and the foundational aspects of mathematics itself, such as set theory and real analysis.