Lebesgue Measure is Diffuse

Theorem
Let $\lambda^n$ be Lebesgue measure on $\R^n$.

Then $\lambda^n$ is a diffuse measure.

Proof
A singleton $\set {\mathbf x} \subseteq \R^n$ is seen to be closed by combining:


 * Euclidean Space is Complete Metric Space
 * Metric Space is Hausdorff
 * Corollary to Compact Subspace of Hausdorff Space is Closed

Hence by Closed Set Measurable in Borel Sigma-Algebra:
 * $\set {\mathbf x} \in \map \BB {\R^n}$

where $\map \BB {\R^n}$ is the Borel $\sigma$-algebra on $\R^n$.

Write $\mathbf x + \epsilon = \tuple {x_1 + \epsilon, \ldots, x_n + \epsilon}$ for $\epsilon > 0$.

Then:


 * $\ds \set {\mathbf x} = \bigcap_{m \mathop \in \N} \horectr {\mathbf x} {\mathbf x + \frac 1 m}$

where $\\horectr {\mathbf x} {\mathbf x + \dfrac 1 m}$ is a half-open $n$-rectangle.

By definition of Lebesgue measure, we have (for all $m \in \N$):


 * $\ds \map {\lambda^n} {\horectr {\mathbf x} {\mathbf x + \frac 1 m} } = \prod_{i \mathop = 1}^n \frac 1 m = m^{-n}$

From Characterization of Measures, it follows that:


 * $\ds \map {\lambda^n} {\set {\mathbf x} } = \lim_{m \mathop \to \infty} m^{-n}$

which equals $0$ from Sequence of Powers of Reciprocals is Null Sequence.

Therefore, for each $\mathbf x \in \R^n$:


 * $\map {\lambda^n} {\set {\mathbf x} } = 0$

that is, $\lambda^n$ is a diffuse measure.