15 - Lebesgue Decomposition, Radon-Nikodym, and Probability Measures
Lecture 15 reviews the Lebesgue decomposition and Radon-Nikodym theorems for $\sigma$-finite measures, works through a mixed measure on $[0,1]$ and the Cantor distribution, links Jacobians to Radon-Nikodym derivatives via change of variables, and refreshes probability basics on random variables, distributions, measurability, and limits.
1. Lebesgue Decomposition Theorem
Given a measurable space $(\Omega,\mathcal{F})$ and σ-finite measures
\(\mu,\ \nu,\)
the Lebesgue decomposition theorem states:
There exist unique measures
\(\nu = \nu_r + \nu_s\) such that:
- $ \nu_r \ll \mu $ (absolutely continuous w.r.t. $ \mu $),
- $ \nu_s \perp \mu $ (singular w.r.t. $ \mu $).
The diagram on page 1 shows a rectangle partitioned into the “regular/AC” part and “singular” part
:contentReference[oaicite:2]{index=2}.
Singular part
\(\nu_s \perp \mu \quad \Longleftrightarrow\quad \exists A\in\mathcal{F}\ \text{such that }\ \nu_s(A^c)=0,\ \mu(A)=0.\)
Absolutely continuous part
(“functions of overlap” in the handwritten note)
\[\nu_r(B) = \int_B g\, d\mu \quad\text{for some measurable } g.\]The function $g$ is the Radon–Nikodym derivative: \(g = \frac{d\nu}{d\mu}.\)
If $ \nu_s = 0$, then $ \nu \ll \mu $.
2. Radon–Nikodym Theorem
Theorem.
If $ \nu \ll \mu$ and $\mu$ is σ-finite, then
\(\exists\ g \ge 0,\ \text{measurable},\quad \nu(B)=\int_B g\, d\mu,\ \forall B\in\mathcal{F}.\)
Your notes reference Appendix A.4 of Durrett.
3. Example 1 (page 1): A Mixed Measure on $[0,1]$
Let:
- $ \Omega=[0,1]$,
- $ \mathcal{F}=\mathcal{B}([0,1])$,
- $ \mu = $ Lebesgue measure,
- $ \delta_{{1/2}} $ = point mass at $1/2$.
Define: \(\nu = \tfrac12 \mu \;+\; \tfrac12 \delta_{\{1/2\}}.\)
Then:
- Absolutely continuous part:
\(\nu_r = \tfrac12 \mu, \qquad \frac{d\nu_r}{d\mu} = \tfrac12.\) - Singular part:
\(\nu_s = \tfrac12 \delta_{\{1/2\}}.\)
Because $\delta_{{1/2}}$ lives entirely on a $\mu$-null set.
4. Example 2 (pages 1–2): Cantor Distribution As Purely Singular
Let ${X_k}_{k\ge1}$ be i.i.d. random variables with:
\[P(X_k = 0) = P(X_k = 2) = \tfrac12.\]Define a random variable $V$ on $[0,1]$ via base 3 expansion:
\[V = \sum_{k=1}^{\infty} \frac{X_k}{3^k}.\]Then:
- $0 \le V \le 1$,
- The distribution of $V$ is supported on the Cantor set $C$,
- $P(V \in C) = 1$, but
- $\mu(C)=0$.
Thus the distribution $\nu$ of $V$ satisfies:
\[\nu_s = \nu,\qquad \nu_r = 0.\]It is purely singular with respect to Lebesgue measure.
Mixed modification
Your notes define:
\[\nu_2 = \tfrac12 \nu + \tfrac12 \mu.\]Then:
- Absolutely continuous part: \(\nu_{2,r} = \tfrac12 \mu,\qquad g = \tfrac12.\)
- Singular part: \(\nu_{2,s} = \tfrac12 \nu.\)
The graph on page 2 shows how adding Lebesgue measure yields a continuous distribution function that still carries Cantor singular mass
:contentReference[oaicite:3]{index=3}.
5. Change of Variables, Jacobian, and RN Derivative
On page 2, the notes draw the diagram of:
\[x = r\cos\theta,\qquad y = r\sin\theta.\]The Jacobian matrix:
\[J = \begin{bmatrix} \frac{\partial x}{\partial \theta} & \frac{\partial x}{\partial r} \\ \frac{\partial y}{\partial \theta} & \frac{\partial y}{\partial r} \end{bmatrix}, \qquad \det J = r = AD - BC.\]The scribbled note says:
“Jacobian = Radon–Nikodym derivative.”
Indeed, in the change-of-variable formula: \(dx\,dy = r\, dr\, d\theta,\) the factor $r$ is the Radon–Nikodym derivative of the pushforward of Lebesgue measure under the transformation.
Formally: \(\frac{d(\mu\circ f^{-1})}{d\lambda} = \vert \det J_f\vert .\)
6. Probability Review (page 2–3)
A probability space:
\[(\Omega,\mathcal{F},P),\qquad P(\Omega)=1.\]A random variable is a measurable map $X:\Omega\to\mathbb{R}$:
\[X^{-1}(B) \in \mathcal{F},\quad B\in\mathcal{B}(\mathbb{R}).\]Distribution of $X$:
\(\mu_X(A) = P(X\in A), \quad A\in\mathcal{B}(\mathbb{R}).\)
If $X$ has density $g$ w.r.t. Lebesgue measure: \(\frac{d\mu_X}{dx} = g.\)
Examples (page 2):
- Uniform$(0,1)$
- Exponential$(\lambda)$
- Normal$(0,1)$, Normal$(\mu,\sigma^2)$.
7. Random Vectors and Measurability (page 3)
A random vector \(\mathbf{X}=(X_1,\ldots,X_d):\Omega\to\mathbb{R}^d\) is measurable iff: \(\mathbf{X}^{-1}(B) \in \mathcal{F}\quad \forall B \in \mathcal{B}(\mathbb{R}^d).\)
To check measurability, it suffices to check rectangles:
\[B = \prod_{i=1}^d (a_i,b_i].\]Function of a random vector
If $f:\mathbb{R}^d \to \mathbb{R}$ is Borel measurable and
$\mathbf{X}$ is measurable, then:
Example (page 3): \(f(x_1,\ldots,x_d)=\sum_{k=1}^d x_k, \quad f(\mathbf{X})=\sum_{k=1}^d X_k.\)
Limits of random variables
If ${X_k}$ are RVs, then:
- $ \inf_k X_k$ is an RV,
- $ \sup_k X_k$ is an RV,
- $ \liminf_k X_k$ and $ \limsup_k X_k$ are RVs.
This uses only closure of measurability under countable operations.
8. Summary of Lecture 15
- Any σ-finite measure decomposes uniquely as
$ \nu = \nu_r + \nu_s $ with $ \nu_r \ll \mu$, $ \nu_s\perp\mu $. - The Radon–Nikodym derivative $g = d\nu/d\mu$ represents densities.
- The Cantor distribution provides a canonical example of a purely singular measure.
- Jacobians in multivariable calculus are Radon–Nikodym derivatives of pushforward measures.
- Random variables and vectors are measurable maps; their distributions arise from pushforward measures.
- Measurability is stable under composition, sums, limits, sup, inf.
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