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Surface densities with vanishing test pairings vanish

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ , the convention of the surface-measure interface cited below (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $Ω \subseteq R^{3}$ be open, let $S \subset Ω$ be a compact embedded $C^{2}$ hypersurface whose surface measure is that of the compact embedded $C^{1}$ class of Surface integration on compact C1 hypersurfaces (the $C^{2}$ data are special cases of the $C^{1}$ data), and let $a : S \to R$ be continuous. Suppose that

$\int_{S} a ψ d S = 0$

for every $ψ \in C_{c}^{\infty} (Ω)$ (Test function space d of an open set), the pairing being the signed surface integral of Surface integration on compact C1 hypersurfaces of the continuous function $a (ψ ∣_{S})$ . For complex tests the integral is defined componentwise in real and imaginary parts; the proof below uses real tests. Then $a = 0$ on $S$ . Equivalently, if two continuous densities on $S$ pair equally against every test function, then they are equal pointwise on $S$ . The conclusion is pointwise and uses the continuity of $a$ ; a merely locally integrable density would satisfy only an almost-everywhere conclusion, which this statement does not claim.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ . An open set $Ω \subseteq R^{3}$ , a compact embedded hypersurface $S \subset Ω$ of the stated class with its surface measure $d S$ , a continuous $a : S \to R$ , and the hypothesis that $\int_{S} a ψ d S = 0$ for every $ψ \in C_{c}^{\infty} (Ω)$ . Subsets of $R^{3}$ carry the subspace metric and the Euclidean distance $d_{2}$ (Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form, Open ball, closed ball and sphere in a metric space).

[F1]

$Ω$ is open: every $x_{0} \in Ω$ has a ball $B (x_{0}, ρ) \subseteq Ω$ (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement). A map is continuous at a point when the usual $ε$ - $δ$ condition holds for the subspace metric (Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form, Open ball, closed ball and sphere in a metric space).

[F2]

The class of Surface integration on compact C1 hypersurfaces consists of finitely many regular injective $C^{1}$ parametrizations $X_{j} : V_{j} \to S \cap O_{j}$ , $O_{j}$ open in $R^{3}$ and $V_{j}$ open in $R^{2}$ , whose images cover $S$ and whose coordinate transitions are $C^{1}$ ; a $C^{1}$ map is continuous ( $C^{k}$ Euclidean maps and diffeomorphisms). For a nonnegative Borel $f$ the integral is the finite chart sum $\int_{S} f d S = \sum_{j} \int_{V_{j}} (χ_{j} f) (X_{j} (y)) J_{X_{j}} (y) d y$ against a subordinate ambient partition, the chart formula restricts to a compact Borel face contained in a regular patch, and a signed $f$ with $\int_{S} ∣ f ∣ d S < \infty$ is integrated through the difference of its positive and negative parts.

[F3]

The chart integral defines a finite Borel measure on $S$ , independent of the finite charts and the subordinate partitions, and every regular chart density $J_{X_{j}}$ is continuous and strictly positive (Chart and partition independence of surface measure).

[F4]

For every $0 < r < R$ and centre $x_{0}$ there is a smooth $ψ : R^{3} \to [0, 1]$ equal to one on ${\overline{B}}_{r} (x_{0})$ with support strictly inside $B_{R} (x_{0})$ (Compactly supported scaled Euclidean bumps); a smooth function whose support is a compact subset of $Ω$ belongs to $D (Ω) = C_{c}^{\infty} (Ω)$ (Test function space d of an open set).

[F5]

The nonnegative Lebesgue integral of a measurable function is monotone and homogeneous for nonnegative scalars, and the integral of the constant multiple $β 1_{C}$ of the indicator of a measurable set $C$ of finite measure is $β λ_{2} (C)$ (The nonnegative Lebesgue integral, Monotonicity and nonnegative homogeneity of the nonnegative integral, The integral of a nonnegative simple function, The nonnegative integral agrees with the simple integral on simple functions).

[F6]

Every closed axis-parallel square $C \subseteq R^{2}$ of side $ℓ > 0$ is Lebesgue measurable with $λ_{2} (C) = ℓ^{2} > 0$ (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

[F7]

The signed Lebesgue integral is defined through the positive and negative parts and is real-linear on integrable functions (Integrable real and complex functions, and their integrals, The Lebesgue integral is linear on $L^{1} (μ)$ ).

[F8]

Continuous real functions on a nonempty compact metric space attain finite extrema (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value), and continuous maps have Borel preimages of Borel sets (A continuous map has Borel preimages of Borel sets). Thus continuous densities on $S$ and compact chart cells are bounded and Borel.

Proof

1.1givenchoosealgebra

Assume that $a (x_{0}) \neq 0$ for some $x_{0} \in S$ and put $m : = ∣ a (x_{0}) ∣ / 2 > 0$ ; let $σ : = sgn (a (x_{0}))$ , so that $σ a (x_{0}) = 2 m > 0$ .

1.2F1choosealgebra

Continuity of $a$ at $x_{0}$ gives $δ_{1} > 0$ with $∣ a (x) - a (x_{0}) ∣ < m$ for every $x \in S$ with $d_{2} (x, x_{0}) < δ_{1}$ , hence $σ a (x) > m$ ; openness of $Ω$ gives $δ_{2} > 0$ with $B (x_{0}, δ_{2}) \subseteq Ω$ . With $N : = B (x_{0}, \min {δ_{1}, δ_{2}})$ we have $N \subseteq Ω$ open and $σ a > m$ on $S \cap N$ .

1.3F2F3F5F8given

For any nonnegative real smooth test $ψ$ , the continuous function $a ψ$ is Borel and bounded on the compact $S$ by [F8], so $\int_{S} ∣ a ψ ∣ d S \leq C S (S) < \infty$ by monotonicity of the nonnegative surface integral and finiteness of the surface measure, and the signed integral $\int_{S} a ψ d S$ is defined; it equals $\int_{S} (a ψ)^{+} d S - \int_{S} (a ψ)^{-} d S$ , and on $S$ we have $(a ψ)^{+} = a^{+} ψ$ and $(a ψ)^{-} = a^{-} ψ$ because $ψ \geq 0$ .

2.1F2step 1.2algebra

The chart family of the class covers $S$ , so $x_{0} \in S \cap O$ for one of its charts $X : V \to S \cap O$ with $V \subseteq R^{2}$ open and $X$ regular, injective and $C^{1}$ ; write $u_{0} : = X^{- 1} (x_{0})$ . Replacing $O$ by $O \cap N$ and $V$ by $X^{- 1} (S \cap (O \cap N))$ , the restricted parametrization is again a regular injective $C^{1}$ chart of the class with $X (V) = S \cap O$ and $O \subseteq N$ , so $σ a > m$ on $X (V)$ .

3.1F4step 2.1choose

Choose $R > 0$ with ${\overline{B}}_{R} (x_{0}) \subseteq O$ and put $r : = R / 2$ ; the scaled Euclidean bump yields $ψ \in C_{c}^{\infty} (R^{3})$ with $0 \leq ψ \leq 1$ , $ψ = 1$ on ${\overline{B}}_{r} (x_{0})$ and $supp ψ \subseteq B_{R} (x_{0}) \subseteq O \subseteq Ω$ , so $ψ \in D (Ω)$ .

4.1F1F2F6step 2.1step 3.1choosealgebra

By continuity of $X$ at $u_{0}$ there is $δ > 0$ with $X [B (u_{0}, δ)] \subseteq B_{r} (x_{0})$ , so choosing $ℓ > 0$ with $ℓ / \sqrt{2} < δ$ and $ℓ$ small enough that the closed axis-parallel square $C : = u_{0} + [- ℓ / 2, ℓ / 2]^{2}$ lies in $V$ gives $C \subseteq B (u_{0}, δ)$ and $X (C) \subseteq B_{r} (x_{0})$ ; then $X (C) \subseteq B_{r} (x_{0}) \subseteq {\overline{B}}_{R} (x_{0}) \subseteq O$ and $λ_{2} (C) = ℓ^{2} > 0$ , and $X (C)$ is a compact Borel face contained in the regular patch $X$ .

5.1F2F3F5F8step 4.1algebra

The chart density $J = J_{X}$ is continuous and strictly positive on $V$ , so [F8] on the nonempty compact square $C$ gives an attained minimum $β : = \min_{C} J > 0$ and $J \geq β$ on $C$ ; since $X$ is injective with $X (V) = S \cap O$ , the restricted chart formula gives $S (X (C)) = \int_{V} 1_{X (C)} (X (y)) J (y) d y = \int_{V} 1_{C} (y) J (y) d y \geq \int_{V} β 1_{C} (y) d y = β λ_{2} (C) = β ℓ^{2} > 0$ .

5.2step 2.1step 3.1step 4.1algebra

On the support of $ψ$ intersected with $S$ the sign $σ$ is constant and $∣ σ a ∣ > m$ ; where $ψ$ vanishes the product $a ψ$ also vanishes. Consequently $σ a ψ \geq m 1_{X (C)}$ pointwise on $S$ : on $X (C)$ we have $ψ = 1$ and $σ a > m$ , and off $X (C)$ the left side is nonnegative on $S \cap supp ψ$ and zero elsewhere.

6.1step 5.1step 1.3step 5.2F2F5givencontradiction

Monotonicity of the nonnegative surface integral, applied through the chart formula, and the evaluation of step 5.1 give $σ \int_{S} a ψ d S = \int_{S} σ a ψ d S \geq m S (X (C)) \geq m β ℓ^{2} > 0$ , so $\int_{S} a ψ d S \neq 0$ ; this contradicts the hypothesis for the test function $ψ \in D (Ω)$ . Hence $a (x_{0}) = 0$ for every $x_{0} \in S$ .

7.1step 6.1F2F3F7F8givenalgebra∎

For the equal-density clause let $a, b : S \to R$ be continuous with $\int_{S} a ψ d S = \int_{S} b ψ d S$ for every $ψ \in C_{c}^{\infty} (Ω)$ ; both $a ψ$ and $b ψ$ are bounded on $S$ , so both signed integrals are defined, and the chartwise representation of the signed integral as a finite sum of signed Lebesgue integrals, together with linearity of the Lebesgue integral, gives $\int_{S} (a - b) ψ d S = \int_{S} a ψ d S - \int_{S} b ψ d S = 0$ for every $ψ$ . The continuous density $c : = a - b$ therefore satisfies the hypothesis of the Statement, and step 6.1 gives $c = 0$ , that is $a = b$ on $S$ .

Source notes

The pointwise conclusion is the standard reduction of a vanishing pairing against every ambient smooth test function to a local positivity argument on the hypersurface. The proof above keeps the library's own chart-measure interface: the graph density of the published treatments appears here only through the strict positivity and continuity of the regular chart density of Chart and partition independence of surface measure, so no product-extension assertion for arbitrary planar tests is used.

Surface densities with vanishing test pairings vanish

Statement

Facts & Assumptions

Proof

Source notes

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