Prove the Set of Points of a Continuous Function is a Borel Set

Borel Function

p (ξ) is a Borel function on Ξ, then μ is a well-defined Borel measure on Ξ. To check that it is invariant for Equation (7.1) we have to verify that

From: Handbook of Dynamical Systems , 2006

Decomposition Theory

In C*-Algebras and their Automorphism Groups (Second Edition), 2018

4.6.11 Lemma

Let $f:T\to E$ be a Borel map from a Polish space T into a second countable, compact space E. Then the graph of f is a Borel subset of $T\times E$ isomorphic with T.

Proof

Define the Borel function $g:T\times E\to E\times E$ by $g(t,s)=(f(t),s)$ . The graph $G(f)$ of f is then the counterimage for g of the diagonal in $E\times E$ . Since the latter is closed, $G(f)$ is a Borel set. By 4.6.10 $G(f)$ is the injective continuous image of a Polish space $T_1$ , and the projection onto the first coordinate gives a continuous bijection of $G(f)$ onto T. By 4.2.10, T, $T_1$ , and $G(f)$ are Borel isomorphic. □

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Preliminaries

Feng-Yu Wang , in Functional Inequalities, Markov Semigroups and Spectral Theory, 2005

Remark 0.3.1

LetF be a complex-valued Borel function on ℝ. Then

exists if and only if

. In particular, ifFis a real-valued continuous function, then (F(L), D(F(L))) defined below is a self-adjoint operator:

$F\left(L\right)f:={\displaystyle {\int }_{-\infty }^{\infty }F\left(\lambda \right)\text{d}{E}_{\lambda }f,}D\left(F\left(\lambda \right)\right):=\left\{{\displaystyle {\int }_{-\infty }^{\infty }F{\left(\lambda \right)}^2\text{d}{\Vert E_{\lambda }f\Vert }^2<\infty }\right\}.$

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Mathematical Statistical Physics

Frank Redig , in Les Houches, 2006

Proposition 4.6

Define

(4.26) ${ℬ}_{\varphi }=\left\{f\in L^1\left(\mu \right):\underset{x\in {\mathbb{Z}}^d}{\Sigma }{\varphi }_x\int \left|a_{x}f-f\right|d\mu <\infty \right\}$

For ϕ satisfying (4.8) all local functions are in ℬ_ϕ. For f ∈ ℬ_ϕ, the expression

(4.27) $Lf=\underset{X\in {\mathbb{Z}}^d}{\Sigma }\left(a_{x}f-f\right)$

is well-defined, i.e., the series converges in L^p (μ) (for all 1 ≤ p ≤ ∞) and moreover,

(4.28) $\underset{t\downarrow 0}{lim}\frac{S\left(t\right)f-f}{t}=Lf$

in L^p(μ).

The following theorem shows that we can start the process from a measure stochastically below μ. We remind this notion briefly here, for more details, see e.g. [24] chapter 2. For η, ξ ∈ Ω we define η ≤ ξ if for all x ∈ ℤ ^d , η _x ≤ ξ _x . Functions f: Ω → ℝ preserving this order are called monotone. Two probability measures ν₁, ν₂ on Ω are ordered (notation ν₁ ≤ ν₂) if for all f monotone, the expectations are ordered, i.e., ν₁ (f) ≤ ν₂ (f). This is equivalent with the existence of a coupling ν₁₂ of ν₁ and ν₂ such that

$v\left\{\left(\eta ,\xi \right):\eta \le \xi \right\}=1$

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Stationary Partial Differential Equations

G. Rozenblum , M. Melgaard , in Handbook of Differential Equations: Stationary Partial Differential Equations, 2005

Theorem A.1

To every self-adjoint operator T there corresponds a unique spectral measure E_T such that

$T={\displaystyle \int s\text{d}{E}_T(s).}$

One has that supp E_T = σ (T).

Having a spectral measure, we can associate to any bounded Borel function f on σ(T) the operator J_f . This mapping f ↦ J_f ≡ ϕ(f) gives the following functional calculus form of the spectral theorem which is very useful.

Theorem A.2

Let T be a self-adjoint operator on ℋ. Then there exists a unique map φ from the bounded Borel functions on σ(T) into ℬ(ℋ) having the following properties:

(i)

ϕ is an algebraic *-homeomorphism, i.e.,

$\begin{array}{ll}\phi (fg)=\phi (f)\phi (g),\hfill & \phi (\lambda f)=\lambda \phi (f),\hfill \\ \phi (1)=I,\hfill & \phi (\overline{f})=\phi {(f)}^{*}.\hfill \end{array}$

(ii)

φ is norm continuous, i.e., $\left|\right|\Psi (f)|{|}_{ℬ\left(\mathscr{H}\right)}⩽|\left|f\right|{|}_{L^{\infty }}$

(iii)

Let {f_n } be a sequence of bounded Borel functions obeying f_n (x) → x for each x (as n → ∞) and |f_n (x)| ⩽ |x| for all x and n. Then, for any $\psi \in D(T),\underset{n\to \infty }{\lim }\psi (f_n)\psi =T\psi$ , lim _n→∞ Φ(f_n )ψ =Tψ.

(iv)

If f_n (x) → f(x) pointwise and if the sequence {‖f_n ‖L ^∞} is bounded, then φ(f_n ) → φ(f) strongly.

(v)

If Tψ = λψ, then φ(f)ψ = f(λ)Ψ.

(vi)

If f ⩾ 0, then φ(f) ⩾ 0.

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Functionals and Representations

In C*-Algebras and their Automorphism Groups (Second Edition), 2018

3.9.7

Dixmier [94] showed that the $C^{⁎}$ -algebra A of bounded Borel functions on $\mathsf{R}$ modulo the ideal of functions vanishing outside a set of first category is a commutative $A{W}^{⁎}$ -algebra (hence a monotone complete $C^{⁎}$ -algebra) with no nonzero completely additive (= normal) functionals. If A had a faithful representation as a von Neumann algebra, then this representation would be normal, since any isomorphism between monotone complete $C^{⁎}$ -algebras is normal (cf. the proof of 2.5.2), but this is impossible by 3.9.3.

The last characterization (by Sakai) of abstract von Neumann algebras is without doubt the most elegant. It is simply the converse of 3.5.6 and helps to understand the profound rôle of the predual of von Neumann algebras (3.6.5).

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Notes on Hyperbolic Systems of Conservation Laws and Transport Equations

Camillo De Lellis , in Handbook of Differential Equations: Evolutionary Equations, 2007

2.7 Alberti's rank-one theorem

In [1] Alberti proved the following deep result.

Theorem 2.13

(Alberti's rank-one theorem). Let B ∈ BV _loc(Ω, ℝ^k ). Then there exist Borel functions ξ : Ω → S ^{d −1},ζ : Ω → S ^k−1 such that

(16) $D^{s}B=\zeta \otimes \xi \left|D^{s}B\right|\mathrm{.}$

Clearly, if we replace D ^s B with D ^j B in (16), this conclusion can be easily drawn from Theorem 2.7. However, in order to prove the same for the full singular part of DB, many new interesting ideas were introduced in [1] (see also [26] for a recent description of Alberti's proof).

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Handbook of Differential Equations

P.I. Plotnikov , J. Sokolowski , in Handbook of Differential Equations: Stationary Partial Differential Equations, 2008

Lemma 6.11.

For a.e. x ∈ Ω, (i) $\mathfrak{M}$ (x, ·) is nonnegative and vanishes on ℝ⁻. Moreover, if the Borel function $\mathfrak{M}$ (x,.) given by (6.20) vanishes d _λ f (x, ·)- almost everywhere on the interval (ω, ∞) with $\omega =\overline{p}{\left(x\right)}^{1/\gamma },$ then d _λ f (x, ·) is a Dirac measure and

$f\left(x,\lambda \right)=0for\lambda <\overline{p}{\left(x\right)}^{1/\gamma },f\left(x,\lambda \right)=1for\lambda \ge \overline{p}{\left(x\right)}^{1/\gamma }.$

(ii)

For all $g\in C_0^{\infty }\left(0,\infty \right),$

(6.34) ${\displaystyle {\int }_{ℝ}g\left(\lambda \right)ℳ\left(x,\lambda \right)d\lambda }=-{\displaystyle {\int }_{\left[0,\infty \right)}{g}^{\prime }\left(\lambda \right){\mathcal{V}}_{\lambda }\left(x\right)d\lambda },$

where $\mathcal{V}$ _λ is defined by (6.29).

Proof. By abuse of notations we will write simply f_k instead of [f], _k . The mollified distribution function f_k (x, ·) belongs to the class C ^∞(ℝ) and generates the absolutely continuous Stieltjes measure σ _kx of the form $d{\sigma }_{kx}={\partial }_{\lambda }{\text{f}}_k{d}_{\lambda }$ . It is easy to see that for k → ∞ the sequence of measures σ _kx converges star-weakly to the measure σ _x = d _λ f in the space of Radon's measures on ℝ. In particular, for all λ with $d_{\lambda }f\left(x,·\right)\left\{\lambda \right\}:={\lim }_{s\to \lambda +0}f\left(x,s\right)-{\lim }_{s\to \lambda -0f}\left(x,s\right)=0$ , we can pass to the limit, to obtain

(6.35) ${\displaystyle {\int }_{\left[0,\lambda \right)}\left(t^{\gamma }-\overline{p}\right){\partial }_t{f}_k\left(x,t\right)dt\to {\displaystyle {\int }_{\left[0,\lambda \right)}\left(t^{\gamma }-\overline{p}\right)d_{t}f\left(x,t\right)}}\text{for}k\to \infty .$

In other words, relation (6.35) holds true for all λ, possibly except for some countable set. Since ${\partial }_{\lambda }{f}_k\ge 0,$ the function on the left-hand side of (6.35) increases on (– ∞, ω) anddecreases on (ω, ∞). From this and (6.35) we conclude that $\mathcal{M}$ (x, ·) does not decrease for λ < ω and does not increase for λ > ω, which along with the obvious relations ${\lim }_{\lambda \to \pm \infty }ℳ\left(x,\lambda \right)=0$ yields the nonnegativity of $\mathcal{M}$ .

In order to prove the second part of (i) note that $\mathfrak{M}\left(x,\lambda \right)={\lim }_{k\to \infty }{\text{S}}_kℳ\left(x,\lambda \right)$ belongs to the first Baire class, and hence is measurable in σ _x . It follows from the monotonicity of $\mathcal{M}$ (x, ·) on the interval (ω, ∞) that if $\mathcal{M}$ (x, α) = 0 for some α > ω, then $\mathcal{M}$ (x, λ) = 0 and f (x, λ) = 1 on (α, ∞). Assume that $\mathcal{M}$ (x, ·) vanishes d _λ f (x, ·)-almost everywhere on (ω, ∞), and consider the set

$\mathcal{O}=\left\{\alpha >\omega :{\sigma }_x\left(\omega ,\alpha \right)\equiv \underset{s\to \alpha -0}{\lim }f\left(x,s\right)-\underset{s\to \omega +0}{\lim }\mathfrak{f}\left(x,s\right)=0\right\}.$

Let us prove that $\mathcal{O}$ = (ω, ∞). If the set $\mathcal{O}$ is empty, then there is a sequence of points ${\lambda }_k\searrow \omega$ with $\mathfrak{M}$ (x, λ _k ) = 0, which yields f (x, ·) = 1 on (ω, ∞) thus $\mathcal{O}$ = (ω, ∞). Hence O ≠ ø. If m = sup O < ∞, then there is a sequence ${\lambda }_k\searrow \omega$ with $\mathfrak{L}$ (x, λ _k ) = 0, which yields f (x, ·) = 1 on (m, ∞). By construction, f (x, λ) = c = constant on (ω, m). In other words, restriction of the Stieltjes measure d _λ f (x, ·) to (ω, ∞) is the mono-atomic measure (1 – c)δ(· – m). Hence $\mathfrak{M}\left(x,m\right)=2^{-1}\left(1-c\right)\left(m^{\gamma }-{\omega }^{\gamma }\right)=0$ which yields c = 1. From this we can conclude that f (x, ·) = 1 on (ω, ∞), and d _λ f (x, ·) is a probability measure concentrated on [0, ω]. Recalling that ω^γ= $\overline{p}$ (x) we obtain

$\mu ℳ\left(x,0\right)={\displaystyle {\int }_{\left[0,\omega \right]}\left({\lambda }^{\gamma }-{\omega }^{\gamma }\right)}{d}_{\lambda }f\left(x,\lambda \right)\ge 0.$

Hence d _λ f (x, λ) is the Dirac measure concentrated at ω, which implies (i).

The proof of (ii) is straightforward. It is easily seen that

$\begin{array}{l}-\mu {\displaystyle {\int }_{ℝ}g\left(\lambda \right)ℳ\left(x,\lambda \right)d\lambda }\\ ={\displaystyle {\int }_{\left[0,\infty \right)}\left({\displaystyle {\int }_{\left[0,\infty \right)}{g}^{\prime }\left(s\right)ds}\right)}\left({\displaystyle {\int }_{\left[0,\infty \right)}\left(t^{\gamma }-\overline{p}\right)d_{t}f\left(x,t\right)}\right)d\lambda \\ ={\displaystyle {\int }_{\left[0,\infty \right)}{g}^{\prime }\left(s\right)\left({\displaystyle {\int }_{\left[0,s\right)}d\lambda }{\displaystyle {\int }_{\left[0,\infty \right)}\left(t^{\gamma }-\overline{p}\right)d_{t}f\left(x,t\right)}\right)}ds\\ ={\displaystyle {\int }_{\left[0,\infty \right)}{g}^{\prime }\left(s\right)\left({\displaystyle {\int }_{\left[0,\infty \right)}\min \left\{t,s\right\}\left(t^{\gamma }-\overline{p}\right)d_t\Gamma \left(x,t\right)}\right)}ds\\ ={\displaystyle {\int }_{\left[0,\infty \right)}{g}^{\prime }\left(s\right)\left(\overline{\min \left\{\varrho ,s\right\}p}-\min \left\{\varrho ,s\right\}\overline{p}\right)ds.}\end{array}$

On the other hand, Condition (H7) of Theorem 6.4 yields $\overline{\min \left\{\varrho ,\lambda \right\}p}-\overline{\min \left\{\varrho ,\lambda \right\}}\overline{p}=\mu {\mathcal{V}}_{\lambda }\left(x\right),$ and the proof of Lemma 6.11 is completed.

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Some Elements of the Classical Measure Theory

Endre Pap , in Handbook of Measure Theory, 2002

4 DEFINITION

A function f: S → $\overline{ℝ}$ is said to be Σ-measurable, if for every Borel set B ⊂ $\overline{ℝ}$ we have $f^{-1}\left(A\right)\in \Sigma$ .

If S is a topological space and Σ = B(S), the Borel σ-algebra of S, a Σ-measurable function f: S → $\overline{ℝ}$ is called a Borel function . Any continuous function f: S → $\overline{ℝ}$ is a Borel function.

EXAMPLES

(a)

Any constant function f: S $\overline{ℝ}$ is Σ-measurable.

(b)

If A ⊂ S, then the characteristic function ${\chi }_A:S\to ℝ$ Σ-measurable if and only if A ∈ Σ.

(c)

Every Σ-step function f: S ∈ ℝ is Σ-measurable.

5 THEOREM

Let f: S → $\overline{ℝ}$ be a Σ-measurable function. Then there is a sequence ${\left(f_n\right)}_{n\in \mathbb{N}}$ of Σ-step functions $f_n:S\to \overline{ℝ}$ such that fn → f pointwise and |fn| ≤ |f| for each n.

If f is positive (with values in [0, + ∞]), the sequence ${\left(f_n\right)}_{n\in \mathbb{N}}$ can be chosen to be increasing.

If f is real-valued and bounded, the sequence ${\left(f_n\right)}_{n\in \mathbb{N}}$ can be chosen to be uniformly convergent.

The following theorem gives a characterization of Σ-measurability.

Σ-measurability is preserved by pointwise convergence:

6 THEOREM

If ${\left(f_n\right)}_{n\in \mathbb{N}}$ is a sequence of $\overline{ℝ}$ valued, Σ-measurable functions converging pointwise to a function f, then f is also Σ-measurable.

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Probabilities and Potential C

In North-Holland Mathematics Studies, 1988

A Method of Compactification

47.

We consider a measure space (E,ε) which we shall suppose Lusin or Souslin (III.16). Let Γbe a convex, ∧-stable cone of bounded, Borel functions on E, containing the positive constants and separating points, separable for uniform convergence (we shall denote by (γ_n) a dense sequence in Γ). We define as in no. 28 the sweeping relation ┤_Γ, the gambling house J …

It is possible to establish a certain number of results on the sweeping relative to Γ, by a reduction to the preceding theory using a compactification procedure, which constitutes a very effective method in potential theory (a method more interesting than what we shall deduce from it here).

a)

Let h be the measurable, injective map × ↦ (γ_n(x))_n∈N from E to ${\text{R}}^{\mathcal{N}}$ ; by III.21 the image h(E) is Borel in ${\mathcal{R}}^{\mathcal{N}}$ (Souslin if E is Soulsin) and h establishes an isomorphism between ε and the Borel σ-field of h(E). So there is no harm in identifying E with its image.

Each function γ_n being bounded, the closure $\overline{\text{E}}$ of E is compact in ${\mathcal{R}}^{\mathcal{N}}$ . As γ_n is identified with the nth coordinate map, it admits a continuous extension to $\overline{\text{E}}$ (unique since E is dense in $\overline{\text{E}}$ ). This property extends by uniform convergence to all f ∈ Γ: we will denote by $\overline{\text{f}}$ the corresponding extension, and by $\overline{\text{Γ}}$ the set of such extensions. It is clear that $\overline{\text{Γ}}$ separates points of $\overline{\text{E}}$ . $\overline{\text{Γ}}$ is a ∧-stable convex cone, which contains the positive constants. By the Stone-Weierstrass theorem $\overline{\text{Γ}}\overline{\text{Γ}}$ is dense in $\text{e}\left(\overline{\text{E}}\right)$ .

Until this last remark we have not used the fact that Γ is a convex cone, nor the stability under ∧.

b)

The gambling house $\overline{\text{J}}$ associated with $\overline{\text{Γ}}$ is compact; the gambling house J is identified with the set of pairs $\left(\text{x,μ}\right)\in \overline{\text{J}}$ such that × ∈ E and μ is carried by E. It is not hard to show (III.60) that J is analytic. We can thus apply the results of §1 to J and $\overline{\text{J}}$ .

Let λ and μ be two positive measures on E, identified to measures on $\overline{\text{E}}$ carried by E). It is clear that the relations $\text{λ}{⊣}_{\overline{\text{Γ}}}\text{μ}$ and $\text{λ}{⊣}_{\text{Γ}}\text{μ}$ are equivalent, and similarly $\text{λ}{⊣}_{{\overline{\text{Γ}}}_{+}}\text{μ}$ and $\text{λ}{⊣}_{{\text{Γ}}_{+}}\text{μ}$ . So, applying 39, we see that $\text{λ}{⊣}_{\text{J}}\text{μ}$ is equivalent to $\text{λ}{⊣}_{\overline{\text{J}}}\text{μ}$ . A little more generally, the restriction to E of each $\overline{\text{J}}$ -sweeping of λ is a J-sweeping of λ.

If h is a $\overline{\text{J}}$ -supermedian function on $\overline{\text{E}}$ , its restriction to E is J-supermedian. Conversely every analytic J-supermedian function f on E is the restriction to E of an analytic $\overline{\text{J}}$ -supermedian function on $\overline{\text{E}}$ : let f⁰ be the extension of f to $\overline{\text{E}}$ which is zero on E^c; as {f⁰ > t} = {f > t} is analytic in E, hence in $\overline{\text{E}}$ , f⁰ is analytic and so is its reduction $\overline{\text{R}}\left({\text{f}}^0\right)$ , and it suffices to show $\overline{\text{R}}\left({\text{f}}^0\right)=\text{f}$ (or simply ≤ f) on E. But, at each point × ∈ E, the left hand side equals ${\text{sup}}_{\text{μ}\in {\overline{\text{J}}}_{\text{x}}}\text{μ}\left({\text{f}}^0\right);$ we can replace μ(f⁰) by η(f), where η is the restriction of μ to E, and as ε_x┤_J η and f is J-supermedian this quantity is indeed majorized by f(x). More generally, if f is analytic we have $\text{R}\text{f}=\overline{\text{R}}\left({\text{f}}^0\right)$ on E.

The compactification method can be used to extend theorem 39: if $\text{λ}{⊣}_{{\text{Γ}}_{\text{+}}}\text{μ}$ there exists ν ≥ μ such that λ┤_Γ ν (it is enough to apply 39 on $\overline{\text{E}}$ and to take a restriction to E).

Another consequence is the extension of Strassen's theorem 40: if λ and μ are carried by E, and λ┤_Γ μ, there exists a kernel $\overline{\text{p}}$ , permitted in $\overline{\text{J}}$ , such that $\text{μ}=\text{λ}\overline{\text{P}}$ . But since $\text{λ}\overline{\text{P}}=\text{μ}$ is carried by E, the set of those × such that ${\text{ε}}_{\text{x}}\overline{\text{P}}$ is not carried by E is λ-negligible; we enclose it in a λ-negligible Borel set B, and we set

${\text{ε}}_{\text{x}}\text{P}={\text{ε}}_{\text{x}}\overline{\text{P}}\text{f}\text{o}\text{r}\text{x}\text{ε}\text{E}\backslash \text{B}{\text{ε}}_{\text{x}}\text{P}={\text{ε}}_{\text{x}}\text{f}\text{o}\text{r}\text{x}\in \text{E}\cap \text{B}$

which gives a permitted kernel P in J such that μ ▭ λP.

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Basic Representation Theory of Groups and Algebras

In Pure and Applied Mathematics, 1988

Proof

Let D ∈ S(M), and let f ∈ L ₊(G). Thus 〈x, m〉 ↦ f(x)Ch _D (xm ) is a bounded Borel function with compact support on G × M; so we can apply Fubini″s Theorem to it, getting:

(5) ${\displaystyle {\int }_M{\displaystyle {\int }_{M}f\left(x\right){\mathrm{Ch}}_D\left(xm\right)d\lambda xd\mu m=}}{\displaystyle {\int }_G{\displaystyle {\int }_{M}f\left(x\right){\mathrm{Ch}}_D\left(xm\right)d\mu md\lambda x.}}$

Now assume that π⁻¹(D) is locally λ-null. Then for all y in G we have (putting π(y) = m):

$\begin{array}{cc}0& ={\displaystyle \int f\left(x{y}^{-1}\right)\Delta \left(y^{-1}\right)C{\text{h}}_{{\pi }^{-1}\left(D\right)}\left(x\right)d\lambda x}\hfill \\ & ={\displaystyle \int f\left(x\right)C{\text{h}}_{{\pi }^{-1}\left(D\right)}\left(xy\right)d\lambda x}\hfill \\ & ={\displaystyle \int f\left(x\right)C{\text{h}}_D\left(xm\right)d\lambda x.}\hfill \end{array}$

So the left side of (5) is 0. Therefore so is the right side; that is, for λ-almost all x,

(6) $0={\displaystyle {\int }_{M}f\left(x\right)C{\text{h}}_D\left(xm\right)d\mu m=f\left(x\right)\mu \left(x^{-1}D\right).}$

Suppose f ≠ 0. It then follows from (6) that μ(x ⁻¹ D) = 0 for some x. Since μ is quasi-invariant, this implies that μ(D) = 0.

Conversely, assume that μ(D) = 0. Then by quasi-invariance μ(x ⁻¹ D) = 0 for all x, so the right side of (5) is 0. Therefore the left side is 0, giving:

(7) $\begin{array}{cc}{\displaystyle \int f\left(x\right)C{\text{h}}_D\left(xm\right)d\lambda x=0}& \text{for}\mu -\text{almost}\text{all}m.\end{array}$

Now let C be a compact subset of G and U a neighborhood of e with compact closure; and suppose f ∈ L ₊(G) is chosen so that f ≥ 1 on CU ⁻¹. Since π is open and the closed support of μ is M (see 14.1), we can choose y in U so that (7) holds for m = π(y). Thus:

(8) $\begin{array}{cc}0& =\Delta \left(y\right){\displaystyle \int f\left(x\right)C{\text{h}}_D\left(\pi \left(xy\right)\right)d\lambda x}\hfill \\ & ={\displaystyle \int f\left(x{y}^{-1}\right)C{\text{h}}_{{\pi }^{-1}\left(D\right)}\left(x\right)d\lambda x.}\hfill \end{array}$

Now if x ∈ C, then xy ⁻¹ ∈ CU ⁻¹. So f(xy ⁻¹)Ch_π⁻¹(D)(x) ≥ Ch_{π⁻¹(D)∩}(x) for all x. Hence by (8) λ(π ⁻¹(D) $\cap$ C) = 0. Since C was an arbitrary compact set, this says that π⁻¹(D) is locally λ-null.

We have now shown that π⁻¹(D) is locally λ-null if and only if μ(D) = 0, and the proof is complete.

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