Multidimensional Integration 2 | The n-dimensional Lebesgue Measure [dark version]

Lebesgue measure in n-dimensional space is built by a single idea: start with measurable sets on the real line, then define the n-dimensional measure on R^n by taking product sets and extending them to a full sigma-algebra. The payoff is a rigorous notion of “length,” “area,” and “volume” in higher dimensions—along with a corresponding n-dimensional Lebesgue integral—without inventing new definitions for each dimension.

In one dimension, the construction begins with a sigma-algebra of Lebesgue-measurable subsets of R, denoted L. Every set A in this sigma-algebra has a well-defined Lebesgue measure Λ(A), taking values in [0, ∞]. This measure generalizes the familiar notion of length on the number line.

To move to two dimensions, the key step is to use the Cartesian product structure of R^2 = R × R. Given two Lebesgue-measurable sets A1 and A2 in R, their product A1 × A2 forms a “generalized rectangle” in R^2. For these product sets, the two-dimensional Lebesgue measure is defined by the product rule: Λ_2(A1 × A2) = Λ(A1) · Λ(A2). From there, the measure is extended to a maximal sigma-algebra of subsets of R^2, called L(R^2), so that many more sets than just rectangles become measurable.

This extension matters because most subsets of R^2 are not simple rectangles, yet the resulting two-dimensional Lebesgue measure still assigns them an “area” in a way consistent with the measure axioms. The resulting sigma-algebra includes all Borel sets (all open and closed sets), and it is complete: any subset of a measure-zero set is also measurable. The measure is normalized so that the unit square has area 1, and it is translation invariant—shifting a measurable set in the plane does not change its measured area.

With Lebesgue measure in place, defining integrals becomes straightforward. The n-dimensional Lebesgue integral is the usual Lebesgue integral taken with respect to the n-dimensional Lebesgue measure. In practice, this means writing an integral over R^n with measure Λ_n, often denoted simply as ∫ f(x) d x when the dimension is clear.

The same product-measure construction repeats in higher dimensions. For R^3, one takes U ⊂ R^2 and A3 ⊂ R and defines volume via Λ_3(U × A3) = Λ_2(U) · Λ(A3). Iterating this process yields the n-dimensional Lebesgue measure Λ_n on R^n, defined on the sigma-algebra L(R^n). It retains the same core properties: completeness, inclusion of Borel sets, normalization on the unit n-cube (measure 1), and translation invariance. The integral in n dimensions follows immediately from measure theory, setting up the next challenge: computing these integrals explicitly when functions and sets are sufficiently “nice.”