Measure Theory 3 | What is a measure? [dark version]

A measure is defined as a function that assigns a generalized “volume” to subsets of a set, but only for subsets belonging to a chosen sigma algebra. The core setup starts with a measurable space: a set X together with a sigma algebra A of subsets of X. A measure is then a map μ from A into the extended nonnegative real numbers—meaning values in [0, ∞] where both 0 and ∞ are allowed. This restriction matters because it guarantees that measured “sizes” never become negative, while still permitting sets of infinite size.

Two rules determine whether such a map deserves the name measure. First, μ(∅) must equal 0, reflecting the idea that the empty set has no volume. Second, μ must be additive over disjoint collections of measurable sets. If A1, A2, …, An are pairwise disjoint and their union is A, then μ(A) equals the sum of μ(Ai). The definition then extends this additivity from finite unions to countably infinite ones: for disjoint measurable sets Ai whose union is A, μ(A) must equal the infinite series Σ μ(Ai). The countable-union requirement is tightly linked to the sigma algebra: since A is closed under countable unions, the union of the Ai stays measurable, so μ can be evaluated.

Once a measure μ is fixed on a sigma algebra A, the triple (X, A, μ) forms a measure space—an environment where “size” can be consistently assigned to the subsets that are deemed measurable. The transcript also emphasizes that the sigma algebra need not be the entire power set of X; in many important settings, only a smaller collection of subsets supports a workable measure.

Several examples ground the definition. The counting measure works on any set X: for a subset A, μ(A) is the number of elements if A is finite, and ∞ if A is infinite. This example highlights how arithmetic with ∞ behaves in measure theory, including the conventions that x + ∞ = ∞ for any x, and x·∞ = ∞ for positive x, while 0·∞ is treated as 0 in many measure-theoretic contexts (though conventions can vary outside the field).

Another example is the Dirac (delta) measure at a point P in X. For any measurable subset A, μ(A) equals 1 if P lies in A and 0 otherwise. Intuitively, all “mass” is concentrated at a single point, like a point charge.

Finally, the discussion points toward the standard Lebesgue measure on R^n, which should generalize ordinary volume. The guiding requirements are: (1) the measure of the unit cube in R^n should be 1, and (2) the measure should be translation invariant—shifting a set by any vector should not change its measured size. The transcript foreshadows that achieving these properties forces a careful choice of sigma algebra (not the full power set), setting up the next steps in measure theory.