Analysis Refresher in the Style of PMA

Ordered fields, completeness, and the real numbers

An ordered field is a field $F$ with a total order $<$ compatible with addition and multiplication:

$$a<b\Rightarrow a+c<b+c,$$

and

$$0<a, \ 0<b\Rightarrow 0<ab.$$

A set $A\subset F$ is bounded above if there exists $M\in F$ such that $a\le M$ for all $a\in A$. A least upper bound, or supremum, is an upper bound $s$ such that $s\le M$ for every upper bound $M$.

The real numbers are characterized as a complete ordered field: every nonempty subset of $\mathbb R$ bounded above has a supremum.

Sequences and limits

A sequence in a set $X$ is a map $\mathbb N\to X$, written $n\mapsto x_n$. In a metric space $(X,d)$, the sequence converges to $x$ if for every $\epsilon>0$ there exists $N$ such that

$$n\ge N\Rightarrow d(x_n,x)<\epsilon.$$

A sequence in a metric space is Cauchy if for every $\epsilon>0$ there exists $N$ such that

$$m,n\ge N\Rightarrow d(x_m,x_n)<\epsilon.$$

Completeness means every Cauchy sequence converges.

For a real sequence $(a_n)$, define

$$\limsup_{n\to\infty}a_n=\lim_{n\to\infty}\sup\{a_k:k\ge n\},$$ $$\liminf_{n\to\infty}a_n=\lim_{n\to\infty}\inf\{a_k:k\ge n\}.$$

The ordinary limit exists if and only if $\limsup a_n=\liminf a_n$.

Numerical series

A series is an expression

$$\sum_{n=1}^{\infty}a_n$$

whose meaning is the limit of partial sums

$$s_N=\sum_{n=1}^{N}a_n.$$

It converges if $(s_N)$ converges.

Absolute convergence means

$$\sum_{n=1}^{\infty}|a_n|<\infty.$$

Absolute convergence implies convergence in $\mathbb R$ or $\mathbb C$.

Useful tests include comparison, ratio, root, alternating series, and Cauchy condensation tests. The conceptual point is that convergence is always about the sequence of partial sums.

Metric spaces and compactness

A metric space is a set $X$ with a distance function $d$. An open ball is

$$B_r(x)=\{y\in X:d(x,y)<r\}.$$

The metric topology is generated by open balls.

A metric space is compact if every open cover has a finite subcover. In metric spaces, compactness is equivalent to sequential compactness:

$$\text{every sequence has a convergent subsequence}.$$

It is also equivalent to completeness plus total boundedness. Total boundedness means that for every $\epsilon>0$, finitely many $\epsilon$-balls cover $X$.

Continuity and uniform continuity

A function $f:X\to Y$ between metric spaces is continuous at $x\in X$ if for every $\epsilon>0$ there exists $\delta>0$ such that

$$d_X(x,x')<\delta\Rightarrow d_Y(f(x),f(x'))<\epsilon.$$

It is continuous if it is continuous at every point.

The topological definition is equivalent:

$$f\text{ is continuous}\quad\Longleftrightarrow\quad f^{-1}(U)\text{ is open for every open }U\subset Y.$$

A function is uniformly continuous if $\delta$ can be chosen independently of $x$.

Connectedness and the intermediate value theorem

A subset of $\mathbb R$ is connected if and only if it is an interval. If $X$ is connected and $f:X\to Y$ is continuous, then $f(X)$ is connected.

The intermediate value theorem follows: if $f:[a,b]\to\mathbb R$ is continuous and $y$ lies between $f(a)$ and $f(b)$, then there exists $c\in[a,b]$ such that

$$f(c)=y.$$

Differentiation in one variable

For $f:(a,b)\to\mathbb R$, the derivative at $x$ is

$$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$

when the limit exists. Differentiability implies continuity.

The mean value theorem states that if $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c\in(a,b)$ such that

$$f(b)-f(a)=f'(c)(b-a).$$

Taylor’s theorem with remainder says that if $f$ has $n+1$ derivatives, then

$$f(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k+R_n(x),$$

where one possible form of the remainder is

$$R_n(x)=\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1}$$

for some $\xi$ between $a$ and $x$.

Riemann and Riemann-Stieltjes integration

Let $f:[a,b]\to\mathbb R$ be bounded. A partition is

$$P=\{a=x_0<x_1<\cdots<x_n=b\}.$$

Upper and lower sums use suprema and infima on subintervals. The Riemann integral exists if the upper and lower integrals agree.

For a monotone increasing function $\alpha:[a,b]\to\mathbb R$, the Riemann-Stieltjes integral

$$\int_a^b f\,d\alpha$$

weights the interval $[x_{i-1},x_i]$ by

$$\alpha(x_i)-\alpha(x_{i-1}).$$

When $\alpha(x)=x$, this is the Riemann integral. When $\alpha$ has jumps, the integral includes point-mass contributions.

Sequences and series of functions

A sequence of functions $f_n:X\to Y$ converges pointwise to $f$ if

$$\lim_{n\to\infty}f_n(x)=f(x)$$

for every $x\in X$. It converges uniformly if for every $\epsilon>0$ there exists $N$ such that

$$n\ge N\Rightarrow d_Y(f_n(x),f(x))<\epsilon$$

for every $x\in X$.

Uniform limits preserve continuity:

$$f_n\text{ continuous and }f_n\to f\text{ uniformly} \quad\Rightarrow\quad f\text{ continuous}.$$

For a series of functions $\sum f_n$, the Weierstrass $M$-test says that if

$$|f_n(x)|\le M_n \qquad\text{for all }x$$

and

$$\sum M_n<\infty,$$

then $\sum f_n$ converges uniformly and absolutely.

Power series and special functions

A power series centered at $a$ is

$$\sum_{n=0}^{\infty}c_n(x-a)^n.$$

There is a radius of convergence $R\in[0,\infty]$ such that the series converges absolutely for $|x-a|<R$ and diverges for $|x-a|>R$. Inside its interval of convergence, a power series can be differentiated and integrated term by term.

The exponential function can be defined by

$$\exp z=\sum_{n=0}^{\infty}\frac{z^n}{n!}.$$

It satisfies

$$\exp(z+w)=\exp z\exp w.$$

The trigonometric functions are encoded by

$$e^{it}=\cos t+i\sin t.$$

Differentiation in several variables

Let $U\subset\mathbb R^n$ be open and $f:U\to\mathbb R^m$. The derivative of $f$ at $a\in U$ is the linear map

$$Df(a):\mathbb R^n\to\mathbb R^m$$

such that

$$f(a+h)=f(a)+Df(a)h+r(h), \qquad \lim_{h\to0}\frac{|r(h)|}{|h|}=0.$$

In coordinates, $Df(a)$ is the Jacobian matrix

$$\left(\frac{\partial f^i}{\partial x^j}(a)\right).$$

The chain rule is

$$D(g\circ f)(a)=Dg(f(a))\circ Df(a).$$

This is the analytic origin of pushforward maps on tangent spaces.

Inverse and implicit function theorems

These theorems justify treating regular level sets as manifolds.

Contraction mapping theorem

A map $T:X\to X$ on a metric space is a contraction if there exists $0<c<1$ such that

$$d(Tx,Ty)\le c\,d(x,y)$$

for all $x,y\in X$.

This theorem is a workhorse behind existence and uniqueness results for ordinary differential equations, including local flows of vector fields.

Ordinary differential equations and flows

Let $X$ be a smooth vector field on an open set $U\subset\mathbb R^n$. An integral curve is a differentiable map

$$\gamma:I\to U$$

satisfying

$$\gamma'(t)=X(\gamma(t)).$$

The local existence and uniqueness theorem for ODEs says that if $X$ is sufficiently smooth, then for every initial point $p\in U$ there is a unique integral curve with $\gamma(0)=p$, at least for small time.

The flow is the map

$$\Phi_t(p)=\gamma_p(t),$$

where $\gamma_p$ is the integral curve starting at $p$. On a manifold, this definition is transported through charts.

Differential forms and Stokes theorem as analysis

On an open subset $U\subset\mathbb R^n$, a $k$-form is a finite sum

$$\omega=\sum_{i_1<\cdots<i_k}f_{i_1\cdots i_k}(x)\,dx^{i_1}\wedge\cdots\wedge dx^{i_k}.$$

If $\sigma:V\subset\mathbb R^k\to U$ parametrizes a $k$-dimensional surface, then the integral of $\omega$ over $\sigma$ is defined by pulling back:

$$\int_\sigma\omega=\int_V \sigma^*\omega.$$

If

$$\sigma^*\omega=g(u)\,du^1\wedge\cdots\wedge du^k,$$

then

$$\int_\sigma\omega=\int_V g(u)\,du^1\cdots du^k.$$

Thus differential forms are algebraic objects first, but integration is defined by pullback to coordinate domains where ordinary multivariable integration applies.

Functions of bounded variation

A function $f:[a,b]\to\mathbb R$ has bounded variation if

$$V_a^b(f)=\sup_P\sum_{j=1}^{n}|f(x_j)-f(x_{j-1})|<\infty,$$

where the supremum is over all partitions $P=\\{a=x_0<\cdots<x_n=b\\}$. Every monotone function has bounded variation. Every function of bounded variation is a difference of two monotone increasing functions.

Bounded variation is the correct regularity class behind Riemann-Stieltjes integration.

Arzela-Ascoli theorem

A family $\mathcal F\subset C(X,Y)$ of functions between metric spaces is equicontinuous if for every $\epsilon>0$ and every $x\in X$, there exists $\delta>0$ such that

$$d_X(x,x')<\delta\Rightarrow d_Y(f(x),f(x'))<\epsilon$$

for all $f\in\mathcal F$.

This theorem is one of the main compactness tools in analysis and differential equations.

Stone-Weierstrass theorem

For $X=[a,b]$, polynomial functions are dense in continuous functions. This justifies approximating complicated functions by algebraically simple ones.

Fourier series

For a $2\pi$-periodic integrable function $f$, define Fourier coefficients

$$\widehat f(n)=\frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-inx}\,dx.$$

The formal Fourier series is

$$\sum_{n\in\mathbb Z}\widehat f(n)e^{inx}.$$

For $f\in L^2(S^1)$, the exponentials form an orthonormal basis and

$$\|f\|_{L^2}^2=2\pi\sum_{n\in\mathbb Z}|\widehat f(n)|^2.$$

Fourier analysis is the analytic background of Bloch theory: periodicity converts translation symmetry into momentum labels.

Differentiation under the integral sign

Let $f(x,t)$ be a function depending on a parameter $t$. A typical rigorous theorem says that if $\partial_t f(x,t)$ exists and is dominated by an integrable function independent of $t$, then

$$\frac{d}{dt}\int f(x,t)\,dx = \int \frac{\partial f}{\partial t}(x,t)\,dx.$$

In modern form, this is usually proved with dominated convergence.

Change of variables

Let $U,V\subset\mathbb R^n$ be open and let $\Phi:U\to V$ be a $C^1$ diffeomorphism. Then for suitable functions $f$,

$$\int_V f(y)\,dy = \int_U f(\Phi(x))\,|\det D\Phi(x)|\,dx.$$

For differential forms, the Jacobian determinant is built into pullback. If

$$\omega=f(y)\,dy^1\wedge\cdots\wedge dy^n,$$

then

$$\Phi^*\omega=f(\Phi(x))\det D\Phi(x)\,dx^1\wedge\cdots\wedge dx^n.$$

The absolute value disappears for oriented form integration because orientation keeps track of the sign.

Lagrange multipliers

Suppose $f,g_1,\ldots,g_m$ are smooth functions on $\mathbb R^n$. To find critical points of $f$ subject to constraints

$$g_1=\cdots=g_m=0,$$

one solves

$$df=\lambda_1 dg_1+\cdots+\lambda_m dg_m$$

provided the differentials $dg_i$ are independent. Geometrically, the derivative of $f$ must vanish on the tangent space to the constraint manifold.

Normed spaces and Banach spaces

A normed vector space is a vector space $V$ with a norm $\|\cdot\|$. It is a Banach space if it is complete in the metric induced by the norm. Examples include $C([a,b])$ with the sup norm and $L^p$ spaces for $1\le p\le\infty$.

The open mapping theorem, closed graph theorem, and uniform boundedness principle are the three basic structural theorems of Banach space theory. They explain why pointwise bounded families of bounded linear operators often have uniform bounds.

Implicit analytic assumptions in geometry

Differential geometry often hides analysis in phrases such as “smooth”, “flow”, and “integrate a form”. The analytic content is:

• smooth maps are locally maps between open subsets of Euclidean spaces with all derivatives,

• tangent maps are derivatives in charts,

• flows are solutions of ODEs,

• form integration is pullback plus ordinary integration,

• Stokes theorem is a far-reaching generalization of the fundamental theorem of calculus.

Measure Theory and Real Analysis Refresher

Sigma-algebras and measurable spaces

Let $X$ be a set. A sigma-algebra $\mathcal A$ on $X$ is a collection of subsets of $X$ such that:

1. $X\in\mathcal A$,

2. if $A\in\mathcal A$, then $X\setminus A\in\mathcal A$,

3. if $A_1,A_2,\ldots\in\mathcal A$, then $\bigcup_{n=1}^{\infty}A_n\in\mathcal A$.

The pair $(X,\mathcal A)$ is a measurable space. Elements of $\mathcal A$ are measurable sets.

If $X$ is a topological space, the Borel sigma-algebra is the smallest sigma-algebra containing all open sets. Its elements are Borel sets.

Measures

A measure on $(X,\mathcal A)$ is a function

$$\mu:\mathcal A\to[0,\infty]$$

such that

$$\mu(\emptyset)=0$$

and for pairwise disjoint measurable sets $A_n$,

$$\mu\left(\bigcup_{n=1}^{\infty}A_n\right)=\sum_{n=1}^{\infty}\mu(A_n).$$

The triple $(X,\mathcal A,\mu)$ is a measure space.

The measure is finite if $\mu(X)<\infty$, sigma-finite if

$$X=\bigcup_{n=1}^{\infty}X_n$$

with $\mu(X_n)<\infty$, and complete if every subset of a measure-zero set is measurable.

Outer measure and Lebesgue measure

An outer measure on $X$ is a function

$$\mu^*:\mathcal P(X)\to[0,\infty]$$

with $\mu^*(\emptyset)=0$, monotonicity, and countable subadditivity.

For $E\subset\mathbb R$, the Lebesgue outer measure is

$$m^*(E)=\inf\left\{\sum_{n=1}^{\infty}(b_n-a_n):E\subset\bigcup_{n=1}^{\infty}(a_n,b_n)\right\}.$$

A set $E\subset\mathbb R$ is Lebesgue measurable if for every $A\subset\mathbb R$,

$$m^*(A)=m^*(A\cap E)+m^*(A\setminus E).$$

The restriction of $m^*$ to measurable sets is Lebesgue measure.

Measurable functions

Let $(X,\mathcal A)$ and $(Y,\mathcal B)$ be measurable spaces. A function $f:X\to Y$ is measurable if

$$f^{-1}(B)\in\mathcal A$$

for every $B\in\mathcal B$.

For real-valued functions, it suffices to check sets of the form

$$\{x:f(x)>a\}$$

for all $a\in\mathbb R$.

A simple function is a finite linear combination of indicator functions:

$$s=\sum_{j=1}^{n}a_j\mathbf 1_{A_j}.$$

Simple functions are the measure-theoretic analogue of step functions.

Lebesgue integration

For a nonnegative simple function

$$s=\sum_{j=1}^{n}a_j\mathbf 1_{A_j}, \qquad a_j\ge0,$$

define

$$\int_X s\,d\mu=\sum_{j=1}^{n}a_j\mu(A_j).$$

For a nonnegative measurable function $f$, define

$$\int_X f\,d\mu=\sup\left\{\int_X s\,d\mu:0\le s\le f, \ s\text{ simple}\right\}.$$

For a general measurable function $f$, write

$$f=f^+-f^- , \qquad f^+=\max(f,0), \qquad f^-=\max(-f,0).$$

If at least one of $\int f^+\,d\mu$ and $\int f^-\,d\mu$ is finite, define

$$\int f\,d\mu=\int f^+\,d\mu-\int f^-\,d\mu.$$

The function is integrable if

$$\int |f|\,d\mu<\infty.$$

Almost everywhere statements

A property holds almost everywhere if it holds outside a measurable set of measure zero. For example,

$$f=g\quad\text{a.e.}$$

means

$$\mu(\{x:f(x)\ne g(x)\})=0.$$

In $L^p$ spaces, functions equal almost everywhere are identified.

Convergence theorems

These theorems are the main reason Lebesgue integration is more powerful than Riemann integration.

Product measures, Tonelli, and Fubini

Given sigma-finite measure spaces $(X,\mathcal A,\mu)$ and $(Y,\mathcal B,\nu)$, there is a product measure

$$\mu\times\nu$$

on the product sigma-algebra, characterized by

$$(\mu\times\nu)(A\times B)=\mu(A)\nu(B).$$

$L^p$ spaces

For $1\le p<\infty$, define

$$\|f\|_p=\left(\int_X |f|^p\,d\mu\right)^{1/p}.$$

The space $L^p(X,\mu)$ consists of equivalence classes of measurable functions with finite $p$-norm. For $p=\infty$,

$$\|f\|_\infty=\operatorname{ess\,sup}|f|.$$

The case $p=2$ is special because

$$\langle f,g\rangle=\int f\overline g\,d\mu$$

makes $L^2$ a Hilbert space.

Signed and complex measures

A signed measure is a countably additive function

$$\nu:\mathcal A\to[-\infty,\infty]$$

that does not take both $+\infty$ and $-\infty$. A complex measure takes values in $\mathbb C$ and is countably additive.

The total variation of a complex measure $\nu$ is

$$|\nu|(E)=\sup\sum_j |\nu(E_j)|,$$

where the supremum is over finite measurable partitions $E=\bigsqcup_jE_j$.

Absolute continuity and Radon-Nikodym theorem

Let $\nu$ and $\mu$ be measures on $(X,\mathcal A)$. We say $\nu$ is absolutely continuous with respect to $\mu$, written

$$\nu\ll\mu,$$

if

$$\mu(E)=0\Rightarrow \nu(E)=0.$$

They are mutually singular, written $\nu\perp\mu$, if there exist disjoint measurable sets $A,B$ with $X=A\cup B$, $\nu$ concentrated on $A$, and $\mu$ concentrated on $B$.

Lebesgue decomposition

This theorem separates a measure into the part described by a density and the singular part.

Weak convergence and distributions of measures

A sequence of finite measures $\mu_n$ on a topological space converges weakly to $\mu$ if

$$\int f\,d\mu_n\to\int f\,d\mu$$

for all bounded continuous test functions $f$ in the chosen class. Weak convergence is central in probability and in functional analysis because it tests measures through observables rather than pointwise densities.

Differentiation of measures and absolute continuity on the line

A function $F:[a,b]\to\mathbb R$ is absolutely continuous if for every $\epsilon>0$ there exists $\delta>0$ such that for any finite collection of disjoint intervals $(a_j,b_j)$,

$$\sum_j(b_j-a_j)<\delta \quad\Rightarrow\quad \sum_j |F(b_j)-F(a_j)|<\epsilon.$$

Approximation and density

On $\mathbb R^n$, smooth compactly supported functions are dense in $L^p$ for $1\le p<\infty$:

$$C_c^\infty(\mathbb R^n)\text{ is dense in }L^p(\mathbb R^n).$$

One standard tool is convolution with a mollifier. A mollifier is a nonnegative smooth compactly supported function $\rho$ with

$$\int_{\mathbb R^n}\rho(x)\,dx=1.$$

Set

$$\rho_\epsilon(x)=\epsilon^{-n}\rho(x/\epsilon).$$

Then $\rho_\epsilon*f$ is smooth and approximates $f$ in suitable senses as $\epsilon\to0$.

Hilbert spaces

A Hilbert space is a complete inner-product space. The norm is

$$\|v\|=\sqrt{\langle v,v\rangle}.$$

Orthogonality, projections, Fourier series, quantum states, and spectral theory are Hilbert-space ideas.

If $M\subset H$ is a closed subspace, then

$$H=M\oplus M^\perp.$$

Bounded linear operators

A linear map $T:X\to Y$ between normed spaces is bounded if there exists $C\ge0$ such that

$$\|Tx\|_Y\le C\|x\|_X$$

for all $x\in X$. Boundedness is equivalent to continuity.

The operator norm is

$$\|T\|=\sup_{\|x\|\le1}\|Tx\|.$$

The bounded operators on a Hilbert space form an algebra. In quantum mechanics, observables are typically represented by self-adjoint operators, often unbounded; handling unbounded operators requires domain care beyond this section.

Distributions and weak derivatives

A test function is usually a smooth compactly supported function. A distribution is a continuous linear functional on a space of test functions. If $u\in L^1_{\mathrm{loc}}(\mathbb R^n)$, it defines a distribution by

$$\varphi\mapsto\int u(x)\varphi(x)\,dx.$$

The weak derivative $D_i u$ is defined by

$$\int (D_i u)\varphi\,dx=-\int u\,\partial_i\varphi\,dx$$

when such a distribution is represented by a function. This idea underlies Sobolev spaces and weak formulations of PDEs.

Measure-theoretic summary for geometry and physics

The main takeaways are:

Caratheodory extension theorem

A premeasure is a countably additive set function defined on an algebra of sets. The Caratheodory extension theorem says that, under standard sigma-finiteness hypotheses, a premeasure extends uniquely to a measure on the generated sigma-algebra.

This is how Lebesgue measure is constructed from interval length and how product measures are constructed from rectangle measures.

Regular Borel measures

On a locally compact Hausdorff space $X$, a Borel measure $\mu$ is inner regular if

$$\mu(E)=\sup\{\mu(K):K\subset E, \ K\text{ compact}\}$$

for suitable measurable sets $E$, and outer regular if

$$\mu(E)=\inf\{\mu(U):E\subset U, \ U\text{ open}\}.$$

Regularity says that measurable sets can be approximated from inside by compact sets and from outside by open sets.

This theorem explains why measures can be studied through integrals of continuous test functions.

Modes of convergence

For measurable functions $f_n,f$:

• $f_n\to f$ pointwise a.e. if $f_n(x)\to f(x)$ outside a null set.

• $f_n\to f$ in measure if for every $\epsilon>0$,

$$\mu(\{x:|f_n(x)-f(x)|>\epsilon\})\to0.$$

• $f_n\to f$ in $L^p$ if

$$\|f_n-f\|_p\to0.$$

For finite measure spaces,

$$L^p\text{ convergence}\Rightarrow\text{ convergence in measure}.$$

A subsequence of a sequence converging in measure converges almost everywhere under suitable hypotheses.

Duality of $L^p$ spaces

Let $1<p<\infty$ and let $q$ be the conjugate exponent, $1/p+1/q=1$. Under sigma-finiteness assumptions,

$$(L^p)^*\cong L^q$$

via

$$g\mapsto\left(f\mapsto\int fg\,d\mu\right).$$

For $p=2$, this identifies $L^2$ with its own dual using the Hilbert inner product.

Hahn-Banach theorem

A key consequence is that normed spaces have enough continuous linear functionals to separate points.

Weak and weak-star topologies

Let $X$ be a normed space. The weak topology on $X$ is the coarsest topology making every continuous linear functional $\ell\in X^*$ continuous. A sequence $x_n$ converges weakly to $x$ if

$$\ell(x_n)\to\ell(x)$$

for every $\ell\in X^*$.

The weak-star topology on $X^*$ is the coarsest topology making evaluation maps

$$\ell\mapsto \ell(x)$$

continuous for every $x\in X$.

This is a fundamental compactness theorem in functional analysis.

Sobolev spaces

For an open set $\Omega\subset\mathbb R^n$, the Sobolev space $W^{k,p}(\Omega)$ consists of functions $u\in L^p(\Omega)$ whose weak derivatives $D^\alpha u$ of order $|\alpha|\le k$ are also in $L^p(\Omega)$. The norm is

$$\|u\|_{W^{k,p}}=\sum_{|\alpha|\le k}\|D^\alpha u\|_{L^p}.$$

For $p=2$, one writes

$$H^k(\Omega)=W^{k,2}(\Omega).$$

Sobolev spaces are the natural setting for PDEs and variational problems.

Fourier transform on $\mathbb R^n$

For a sufficiently nice function $f:\mathbb R^n\to\mathbb C$, define

$$\widehat f(\xi)=\int_{\mathbb R^n}e^{-ix\cdot\xi}f(x)\,dx.$$

With normalization conventions adjusted appropriately, the inverse transform is

$$f(x)=\frac{1}{(2\pi)^n}\int_{\mathbb R^n}e^{ix\cdot\xi}\widehat f(\xi)\,d\xi.$$

Differentiation becomes multiplication:

$$\widehat{\partial_j f}(\xi)=i\xi_j\widehat f(\xi).$$

This is why Fourier analysis diagonalizes constant-coefficient differential operators and why momentum space is natural in band theory.

Spectral theorem, bounded self-adjoint case

Let $H$ be a Hilbert space and $T:H\to H$ a bounded self-adjoint operator. The spectral theorem says that $T$ can be represented as multiplication by a real-valued function on an $L^2$ space, or equivalently through a projection-valued measure:

$$T=\int_\mathbb R \lambda\,dE(\lambda).$$

For finite-dimensional Hermitian matrices, this reduces to diagonalization by a unitary matrix. In quantum mechanics, the spectral theorem is the rigorous foundation for observables and measurement probabilities.

Probability language

A probability space is a measure space $(\Omega,\mathcal F,\mathbb P)$ with $\mathbb P(\Omega)=1$. A random variable is a measurable function

$$X:\Omega\to\mathbb R.$$

Expectation is integration:

$$\mathbb E[X]=\int_\Omega X\,d\mathbb P.$$

Independence of sigma-algebras or random variables is a multiplicativity condition on probabilities. Many statistical mechanics constructions can be phrased measure-theoretically, even when physicists use more formal path-integral notation.

What this section deliberately does not hide

Measure theory separates three logically different operations:

1. choosing measurable sets,

2. assigning size to those sets,

3. integrating measurable functions against that size assignment.

Differential forms add geometry and orientation to integration. Measures add limit stability and functional analysis. Both are needed in modern mathematical physics, but they solve different problems.