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One-dimensional substitution on a deflection branch: the outgoing angular measure has density $b ∣ d b / d θ ∣$ , hence cross-section density $b ∣ d b / d θ ∣ / \sin θ$

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $θ : (a_{1}, a_{2}) \to (θ_{1}, θ_{2}) \subseteq (0, π)$ be a deflection branch of Scattering data: the incident flux measure, the solid-angle element, deflection branches and the cross-section density, with inverse $b (θ)$ , and set $D = (a_{1}, a_{2}) \times (0, 2 π)$ , $H = (θ_{1}, θ_{2}) \times (0, 2 π)$ and $T (b, φ) = (θ (b), φ)$ . Restrict the incident measure $ν$ to $D$ . Its pushforward, defined by $T_{*} ν (B) = ν (T^{- 1} (B))$ for Borel $B \subseteq H$ , satisfies $T_{*} ν (B) = \int_{B} b (θ) ∣ b^{'} (θ) ∣ d λ_{2} (θ, φ) .$ For every nonnegative Borel $g : H \to [0, + \infty]$ , and for every real Borel $g$ with either side's integral of $∣ g ∣$ finite, consequently $\int_{D} g (θ (b), φ) b d λ_{2} (b, φ) = \int_{H} g (θ, φ) b (θ) ∣ b^{'} (θ) ∣ d λ_{2} (θ, φ) .$ The nonnegative equality allows $+ \infty$ ; in the real case both sides are finite and absolute integrability on one side is equivalent to that on the other. In particular it applies to continuous periodic test functions that are nonnegative or absolutely integrable, and to compactly supported continuous test functions. No equality of undefined signed improper integrals is asserted. Nonnegative double-integral notation denotes the corresponding Lebesgue integrals, with iterated integrals justified by Tonelli.

Thus the outgoing angular measure has density $b (θ) ∣ b^{'} (θ) ∣$ against $d θ d φ$ and density $\frac{d σ}{d Ω} (θ) = \frac{b (θ) ∣ b^{'} (θ) ∣}{\sin θ} = \frac{b (θ)}{\sin θ ∣ θ^{'} (b (θ)) ∣}$ against the solid-angle measure. For finitely many branches on disjoint incident intervals the measures add, including on overlapping image intervals; in particular contributions with pairwise disjoint images add.

Facts & Assumptions

Given: Countable choice, the branch, its inverse, $D, H, T, ν$ and the test function as stated.

[L1]

The inverse is $C^{1}$ , with $b^{'} = 1 / (θ^{'} \circ b)$ ; the sign of the continuous nonzero derivative is constant, so the branch is strictly monotone (Derivative of an inverse: if $f$ is continuous and injective on a nondegenerate interval $I$ and differentiable at $c \in I$ with $f^{'} (c) \neq 0$ , then the inverse $g$ is differentiable at $f (c)$ with $g^{'} (f (c)) = 1 / f^{'} (c)$ ; and if $f^{'} (c) = 0$ then $g$ is not differentiable at $f (c)$ , The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ , Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point).

[L2]

Compact substitution with absolute derivative holds on every closed interval lying strictly inside the branch; continuous functions there are Riemann integrable and their Riemann and Lebesgue integrals agree under countable choice (In one dimension the compact-Jordan formula is substitution over the unoriented image interval with the absolute derivative, A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral).

[L3]

On Borel subsets of the plane $λ_{2} = λ_{1} \times λ_{1}$ , and Tonelli applies to nonnegative Borel integrands. Measures agreeing on a generating pi-system with an increasing common finite-measure exhaustion agree everywhere (On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}, Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Measures agreeing on a generating pi-system are equal under an increasing finite-measure exhaustion from that pi-system).

[L4]

Continuous maps have Borel preimages; a nonnegative measurable density defines a measure. Nonnegative measurable functions have increasing simple approximations and monotone convergence holds; real integrability means finite integral of the absolute value (A continuous map has Borel preimages of Borel sets, The indefinite integral of a nonnegative measurable function is a measure, Every nonnegative measurable function is the increasing limit of simple measurable functions, Monotone convergence for the integral, Integrable real and complex functions, and their integrals, Scattering data: the incident flux measure, the solid-angle element, deflection branches and the cross-section density).

[L5]

$\sin θ > 0$ on $(0, π)$ : the solid-angle definition proves this from $\sin 0 = \sin π = 0$ and strict increase/decrease on the two half intervals; its continuous density is Borel measurable (Scattering data: the incident flux measure, the solid-angle element, deflection branches and the cross-section density).

Proof

technique · direct

1.1givenL1L4

By [L1], $b$ is $C^{1}$ and $h (θ) = b (θ) ∣ b^{'} (θ) ∣$ is continuous and positive. The map $T$ is a homeomorphism $D \to H$ , so $T_{*} ν$ is a Borel measure: preimages are Borel and preserve disjoint countable unions. The nonnegative density $h$ defines another Borel measure $η (B) = \int_{B} h d λ_{2}$ by [L4].

2.1step 1.1L2L3algebra

Let $c < d$ lie in $(θ_{1}, θ_{2})$ and $0 < p < q < 2 π$ . Apply compact substitution [L2] to the inverse map $b$ on $[c, d]$ , with integrand $x \mapsto x$ : writing $u = \min (b (c), b (d))$ , $v = \max (b (c), b (d))$ , it gives $\int_{c}^{d} h (θ) d θ = \int_{u}^{v} x d x$ . Endpoints have Lebesgue measure zero, so Tonelli [L3] yields $η ((c, d) \times (p, q)) = (q - p) \int_{u}^{v} x d x = ν (T^{- 1} ((c, d) \times (p, q)))$ . These quantities are finite.

3.1step 1.1step 2.1L3

Interior open rectangles, together with the empty set, form a pi-system generating the Borel sets of $H$ : rational rectangles form a countable base, and every open set is the union of those base rectangles it contains. Choose explicit nested interior intervals exhausting each open coordinate interval; their product rectangles exhaust $H$ and have the common finite measure of step 2.1. Thus measure uniqueness [L3] gives $T_{*} ν = η$ on every Borel set of $H$ , proving the asserted pushforward identity even for unbounded open branches.

4.1step 3.1L3L4algebra

The test-function identity first holds for indicator functions by step 3.1 and the definition of pushforward, then for nonnegative simple functions by finite addition, and finally for every nonnegative Borel $g$ by increasing simple approximation and monotone convergence [L4] on both sides. Apply this to $∣ g ∣$ to obtain equivalence of absolute integrability; applying it to the positive and negative parts gives the real identity whenever that absolute integral is finite. Tonelli [L3] supplies the nonnegative iterated form.

5.1step 1.1step 3.1step 4.1L1L4L5algebra∎

Since $\sin θ > 0$ by [L5], rewriting $h d λ_{2} = (h / \sin θ) d Ω$ gives the solid-angle density; [L1] gives its reciprocal-derivative expression. Pushforward commutes with finite addition of measures by its definition, so branches on disjoint incident intervals contribute the sum of these measures, and the sum of the densities wherever their images overlap. This proves all assertions.

One-dimensional substitution on a deflection branch: the outgoing angular measure has density $b ∣ d b / d θ ∣$ , hence cross-section density $b ∣ d b / d θ ∣ / \sin θ$

Statement

Facts & Assumptions

Proof

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Dependency tree · 2 levels

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One-dimensional substitution on a deflection branch: the outgoing angular measure has density b∣db/dθ∣b∣db/dθ∣, hence cross-section density b∣db/dθ∣/sin⁡θb∣db/dθ∣/sinθ

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

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One-dimensional substitution on a deflection branch: the outgoing angular measure has density $b ∣ d b / d θ ∣$ , hence cross-section density $b ∣ d b / d θ ∣ / \sin θ$