Differentiation of Measures - Examples and Constructions

Understanding the differentiation of measures through concrete examples illuminates the theory and its applications to analysis and probability.

ExampleIndefinite Integral Construction

Let $f(x) = x$ on $[0, 1]$ . Then $f \in L^1[0, 1]$ and: $F(x) = \int_0^x t \, dt = \frac{x^2}{2}$

We verify:

$F$ is absolutely continuous (being continuously differentiable)
$F'(x) = x = f(x)$ for all $x$ (not just a.e.)
$F(1) - F(0) = \int_0^1 F'(t) \, dt = \int_0^1 t \, dt = \frac{1}{2}$

This confirms the Fundamental Theorem.

ExampleFunctions of Bounded Variation Decomposition

Let $F(x) = x + [\![x]\!]$ where $[\![x]\!]$ is the greatest integer function. On $[0, 3]$ :

$F$ has jumps at $x = 1, 2, 3$ with $F$ increasing by $2$ at each jump
$F$ increases by $1$ on each unit interval
Total variation: $V_0^3(F) = 3 \cdot 1 + 3 \cdot 1 = 6$

We can decompose $F = F_c + F_j$ where:

$F_c(x) = x$ is continuous
$F_j(x) = [\![x]\!]$ is a jump function

This illustrates how BV functions decompose into continuous and discrete parts.

ExampleSingular Continuous Functions

The Cantor-Lebesgue function $C: [0, 1] \to [0, 1]$ is constructed as follows:

On the middle third $(\frac{1}{3}, \frac{2}{3})$ , set $C = \frac{1}{2}$
On each removed interval at stage $n$ , $C$ takes a constant value (dyadic rationals)
Extend by continuity to the Cantor set

Properties:

$C$ is continuous and increasing
$C(0) = 0$ , $C(1) = 1$
$C' = 0$ almost everywhere (on the complement of the Cantor set, which has measure $1$ )
$C$ is NOT absolutely continuous

This function shows that continuous, increasing functions need not be absolutely continuous, and derivatives can vanish a.e. even when the function increases.

Remark

Construction techniques and applications:

Vitali covering lemma: Essential for proving differentiation theorems. It states that from any collection of intervals covering a set in a "fine" way, one can extract a countable disjoint subcollection covering most of the set.
Maximal functions: The Hardy-Littlewood maximal function: $Mf(x) = \sup_{h > 0} \frac{1}{2h} \int_{x-h}^{x+h} |f(t)| \, dt$

is used to prove differentiation theorems, showing that for $f \in L^1_{\text{loc}}$ : $\lim_{h \to 0} \frac{1}{2h} \int_{x-h}^{x+h} f(t) \, dt = f(x) \text{ a.e.}$

Radon-Nikodym in differentiation: The derivative $\frac{d\nu}{d\mu}$ can be interpreted as a limit: $\frac{d\nu}{d\mu}(x) = \lim_{r \to 0} \frac{\nu(B_r(x))}{\mu(B_r(x))}$

where $B_r(x)$ are balls shrinking to $x$ .

Signed measures from BV functions: Every function of bounded variation $F$ generates a signed measure $\mu_F$ via: $\mu_F([a, b]) = F(b) - F(a)$

The Radon-Nikodym derivative $\frac{d\mu_F}{dx}$ equals $F'(x)$ almost everywhere when $F$ is absolutely continuous.

These examples demonstrate the rich structure of the theory of differentiation in measure theory, connecting classical analysis with modern integration theory.