Measure Theory: Foundations¶

Job: construct and manipulate measures; integrate; pass limits under $\int$; factor product measures and densities; represent duals of $C_c$ / $C_0$ (Riesz).

Core¶

Measurable spaces and measures¶

A $\sigma$-algebra $\mathcal{M}$ on $X$: contains $X$, closed under complements and countable unions. Measure $\mu:\mathcal{M}\to[0,\infty]$: $\mu(\emptyset)=0$, countable additivity on disjoint unions. Finite / $\sigma$-finite / probability as usual.

Borel $\sigma$-algebra $\mathcal{B}(X)$: generated by opens. Lebesgue $\sigma$-algebra on $\mathbb{R}^n$: completion of Borel under Lebesgue measure.

Outer measure and Carathéodory¶

Outer measure $\mu^*$: monotone, countably subadditive, $\mu^*(\emptyset)=0$. Carathéodory measurable: $E$ such that $\forall A$, $$ \mu^(A)=\mu^(A\cap E)+\mu^*(A\cap E^c). $$

Theorem (Carathéodory). Measurable sets form a $\sigma$-algebra; $\mu^*|_{\mathcal{M}}$ is a complete measure.

Lebesgue measure: start from length on intervals, generate outer measure, restrict to Carathéodory / complete Borel. Translation invariant; unique on Borel among locally finite translation-invariant measures up to scalar (Haar uniqueness on $\mathbb{R}^n$).

Measurable functions and integration¶

$f:X\to\overline{\mathbb{R}}$ measurable: $\{f>a\}\in\mathcal{M}$ (equiv. forms). Approximation by simple functions: $0\le f_n\uparrow f$ for $f\ge 0$ measurable.

$$ \int f\,d\mu:=\sup\sum a_i\mu(E_i)\quad\text{(simple)};\quad\text{extend by monotone limits}. $$ Integrable: $\int|f|<\infty$. Absolute integrability $\Leftrightarrow$ integral defined and finite in $\mathbb{R}$.

Limit theorems¶

Theorem	Hypotheses $\Rightarrow$ conclusion
Monotone convergence (MCT)	$0\le f_n\uparrow f$ $\Rightarrow$ $\int f_n\to\int f$
Fatou	$f_n\ge 0$ $\Rightarrow$ $\int\liminf f_n\le\liminf\int f_n$
Dominated convergence (DCT)	$f_n\to f$ a.e., $\\|f_n\\|\le g\in L^1$ $\Rightarrow$ $\int f_n\to\int f$, $L^1$ conv.
Bounded convergence	finite measure + unif. bound + pointwise $\Rightarrow$ $L^1$ (special DCT)
Vitali (prob.)	UI + conv. in prob. $\Rightarrow$ $L^1$ conv. (finite measure)

Product measures and Fubini (detail)¶

Product $\sigma$-algebra $\mathcal{M}\otimes\mathcal{N}$: generated by measurable rectangles $A\times B$. For $\sigma$-finite $(X,\mu)$, $(Y,\nu)$, there is a unique measure $\mu\otimes\nu$ on $\mathcal{M}\otimes\mathcal{N}$ with $$ (\mu\otimes\nu)(A\times B)=\mu(A)\nu(B). $$ Construction: for $E\subset X\times Y$ measurable, sections $E_x=\{y:(x,y)\in E\}$ are measurable for each $x$, and $x\mapsto\nu(E_x)$ is measurable, with $(\mu\otimes\nu)(E)=\int \nu(E_x)\,d\mu(x)$ (and symmetrically). Completion of $\mu\otimes\nu$ is needed for full Lebesgue theory on $\mathbb{R}^{m+n}$; Fubini on the completion requires care with null-section pathologies.

Theorem (Tonelli). $f\ge 0$ measurable on product ($\sigma$-finite factors):
$\int f\,d(\mu\otimes\nu)=\int\bigl(\int f(x,y)\,d\nu(y)\bigr)d\mu(x)$ (either order; both sides in $[0,\infty]$).

Theorem (Fubini). $f\in L^1(\mu\otimes\nu)$: same equality with finite integrals; for a.e. $x$, $y\mapsto f(x,y)$ is integrable, and $x\mapsto\int f(x,y)\,d\nu$ is integrable (symmetrically).

Infinite products: for probability measures, Kolmogorov extension builds consistent processes on $\prod X_i$ from consistent finite-dimensional distributions (topological hypotheses: Polish spaces).

Absolute continuity and Radon–Nikodym¶

$\nu\ll\mu$ (on $\sigma$-finite setup): $\mu(E)=0\Rightarrow\nu(E)=0$.

Theorem (Radon–Nikodym). $\nu\ll\mu$, both $\sigma$-finite $\Rightarrow$ $\exists$ measurable $f\ge 0$ with $\nu(E)=\int_E f\,d\mu$. Write $f=d\nu/d\mu$. Unique a.e.

Lebesgue decomposition: $\nu=\nu_{\mathrm{ac}}+\nu_{\mathrm{s}}$ with $\nu_{\mathrm{ac}}\ll\mu$, $\nu_{\mathrm{s}}\perp\mu$. Hahn–Jordan: signed measures decompose into positive/negative parts.

Corollary ($L^p$ duals). For $1\le p<\infty$ and $\sigma$-finite $\mu$: $(L^p)^*\cong L^q$ via $g\mapsto\int fg$. For $p=\infty$, dual is larger than $L^1$ in general (finitely additive measures).

Riesz representation¶

Theorem (Riesz–Markov–Kakutani, compact version). $X$ compact Hausdorff. Positive linear functionals $\Lambda:C(X)\to\mathbb{R}$ correspond to unique regular Borel measures $\mu$ with $\Lambda(f)=\int f\,d\mu$. Bounded linear functionals on $C(X)$ $\leftrightarrow$ regular signed/complex Borel measures (total variation norm).

Locally compact version. $C_c(X)$ dual $\leftrightarrow$ Radon measures; $C_0(X)^*$ $\leftrightarrow$ finite regular Borel measures. This is the measure-theoretic engine for duals of continuous-function spaces and for defining distributions of continuous maps (pushforwards).

Haar measure (pointer). Locally compact groups carry nonzero left-invariant Radon measures, unique up to scalar — Riesz applied to invariant functionals.

Differentiation of measures (pointer)¶

Vitali / Besicovitch covering $\Rightarrow$ Lebesgue differentiation: averages of integrable $f$ recover $f(x)$ a.e. Hardy–Littlewood maximal function controls weak-type $(1,1)$. Radon–Nikodym derivatives as pointwise limits of $\nu(B_r)/\mu(B_r)$ under doubling/hypotheses.

$L^p$ quick structure¶

$\|f\|_p=(\int|f|^p)^{1/p}$; Minkowski, Hölder. $1\le p<\infty$: separable when underlying measure is $\sigma$-finite on a countably generated $\sigma$-algebra (standard spaces). Completeness: $L^p$ Banach; $L^2$ Hilbert.

See it¶

Change of variables / pushforwards. If $T:X\to Y$ measurable, $(T_*\mu)(B)=\mu(T^{-1}B)$. Densities transform by Jacobian when $T$ is $C^1$ diffeomorphism on $\mathbb{R}^n$: classical change of variables is RN for Lebesgue under $T$.

Kakutani / infinite product measures. Product of probability measures on infinite products; absolute continuity criteria for Gaussian measures on Banach spaces — RN in infinite dimensions.

Conditional expectation. $\mathbb{E}[X\mid\mathcal{G}]$ is the RN derivative of $A\mapsto\int_A X\,d\mathbb{P}$ w.r.t. $\mathbb{P}|_{\mathcal{G}}$ on $L^1$ — uniqueness only a.s.

Check¶

Sketch why DCT fails without domination: the sliding bump.

On $[0,1]$, $f_n=n1_{[0,1/n]}$: $\int f_n=1\not\to 0=\int\lim f_n$. No single integrable $g\ge f_n$ all $n$.

If $\nu\perp\mu$ and $\nu\ll\mu$, conclude $\nu=0$. Why?

$\exists E$ with $\mu(E)=0$ and $\nu(E^c)=0$. Then $\nu\ll\mu\Rightarrow\nu(E)=0$, so $\nu(X)=0$.

Why does Fubini need $\sigma$-finiteness (or care)?

Without $\sigma$-finiteness, product measures may fail uniqueness or section measurability pathologies appear; standard counterexamples use non-$\sigma$-finite counting measure on uncountable sets.

Pitfalls¶

Non-measurable sets exist via AC (Vitali); explicit analysis sets are Borel/Lebesgue.
Tonelli needs nonnegativity or integrability — without either, iterated integrals may disagree.
Product $\sigma$-algebra vs completion — null sets in each variable need the completed product.
$L^p$ inclusions depend on finite vs infinite measure.
Riesz measures are regular; not every abstract measure on Borel $\sigma$-algebra is Radon on non-nice spaces.

Refs¶

Folland, Real Analysis: Modern Techniques and Their Applications
Rudin, Real and Complex Analysis
Cohn, Measure Theory
Bogachev, Measure Theory

Next¶

MT index · Geometric Measure Theory · Ergodic & Recurrence · Probability & Stochastic · Harmonic & Singular Integrals · Functional analysis · Probability · Real analysis · Fourier analysis · Harmonic analysis — $L^p$ spaces, maximal functions, CZ singular integrals