LemmaStatement: AI-adaptedProof: AI-adaptedprecheck pass

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Component amplitudes, coherent and incoherent bin laws, and recorded detector weights

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), the choice hypothesis of the cited Hilbert-space and measure suppliers. Let $(D, A, d x)$ be a $σ$ -finite measure space (Measures on sigma-algebras), let $r \geq 1$ be an integer, and let $L^{2} (D; C^{r}) : = {f : D \to C^{r} measurable : \int_{D} ∥ f (x) ∥_{C^{r}}^{2} d x < \infty} / ⁣ \sim,$ the quotient by equality $d x$ -almost everywhere (The space $L^{p} (μ)$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions), with $⟨ f, g ⟩ : = \int_{D} ⟨ f (x), g (x) ⟩_{C^{r}} d x, ∥ f ∥^{2} : = ⟨ f, f ⟩ = \int_{D} ∥ f (x) ∥_{C^{r}}^{2} d x,$ linear in the first argument ( $L^{2}$ with the integral pairing is a Hilbert space). For $a, b \in L^{2} (D; C^{r})$ and measurable $B \subseteq D$ write $⟨ a, b ⟩_{B} : = \int_{B} ⟨ a (x), b (x) ⟩_{C^{r}} d x, N : = ∥ a ∥^{2} + ∥ b ∥^{2}, c : = Re ⟨ a, b ⟩_{D} .$ Then:

(a) Pointwise and integrated cross term. For $d x$ -almost every $x \in D$ , $∥ a (x) + b (x) ∥_{C^{r}}^{2} = ∥ a (x) ∥_{C^{r}}^{2} + ∥ b (x) ∥_{C^{r}}^{2} + 2 Re ⟨ a (x), b (x) ⟩_{C^{r}},$ and for every measurable $B \subseteq D$ , the three functions $∥ a ∥^{2}$ , $∥ b ∥^{2}$ , $⟨ a, b ⟩_{C^{r}}$ being $d x$ -integrable on $B$ , $\int_{B} ∥ a + b ∥^{2} d x = \int_{B} ∥ a ∥^{2} d x + \int_{B} ∥ b ∥^{2} d x + 2 Re ⟨ a, b ⟩_{B} .$

(b) Coherent bin law. If $∥ a + b ∥ > 0$ , then $p_{c o h} (B) : = \frac{\int_{B} ∥ a + b ∥^{2} d x}{∥ a + b ∥^{2}} (B \in A)$ defines a probability measure on $(D, A)$ (Probability measures and probability spaces), and for every $B \in A$ the expansion of (a) holds with $∥ a + b ∥^{2} = N + 2 c$ in the denominator.

(c) Incoherent bin law. If $∥ a ∥ > 0$ , $∥ b ∥ > 0$ and $w \in [0, 1]$ , then $p_{m i x} (B) : = w \frac{\int_{B} ∥ a ∥^{2} d x}{∥ a ∥^{2}} + (1 - w) \frac{\int_{B} ∥ b ∥^{2} d x}{∥ b ∥^{2}} (B \in A)$ defines a probability measure on $(D, A)$ .

(d) Recorded weights, sub-probability and conditioning. Let $μ$ be a probability measure on $(D, A)$ and let $W : D \to [0, 1]$ be measurable. Then $B \mapsto ν (B) : = \int_{B} W d μ$ is a finite measure on $(D, A)$ with $0 \leq ν (D) \leq 1$ ; if $ν (D) > 0$ , then $\overset{ˉ}{ν} (B) : = ν (B) / ν (D)$ defines a probability measure on $(D, A)$ . More generally, for finitely many measurable weights $W_{1}, \dots, W_{m} : D \to [0, 1]$ with $\sum_{a = 1}^{m} W_{a} \leq 1$ pointwise, the numbers $ν_{a} : = \int_{D} W_{a} d μ$ satisfy $ν_{a} \geq 0$ and $\sum_{a = 1}^{m} ν_{a} \leq 1$ ; if $\sum_{a = 1}^{m} W_{a} = 1$ pointwise, then $\sum_{a = 1}^{m} ν_{a} = 1$ .

(e) Coincidence criterion. Let $∥ a ∥ > 0$ , $∥ b ∥ > 0$ and $∥ a + b ∥ > 0$ , and put $w_{*} : = ∥ a ∥^{2} / N$ , so that $p_{m i x}$ with weight $w_{*}$ is $p_{*} (B) = N^{- 1} \int_{B} (∥ a ∥^{2} + ∥ b ∥^{2}) d x$ . Then $p_{c o h} (B) = p_{*} (B)$ for every $B \in A$ if and only if $N Re ⟨ a (x), b (x) ⟩_{C^{r}} = c (∥ a (x) ∥_{C^{r}}^{2} + ∥ b (x) ∥_{C^{r}}^{2}) for d x -almost every x \in D .$ In particular, if $⟨ a (x), b (x) ⟩_{C^{r}} = 0$ for almost every $x$ , then the two laws coincide; and if $c = 0$ , then they coincide if and only if $Re ⟨ a (x), b (x) ⟩_{C^{r}} = 0$ for almost every $x$ .

The assertions concern the specified classes and weights only; no probability law of an apparatus, no sample of trials and no empirical claim is asserted.

Facts & Assumptions

Given: The Axiom of Countable Choice; a $σ$ -finite measure space $(D, A, d x)$ ; an integer $r \geq 1$ ; classes $a, b \in L^{2} (D; C^{r})$ ; a probability measure $μ$ on $(D, A)$ ; measurable weights $W, W_{1}, \dots, W_{m} : D \to [0, 1]$ ; reals $w \in [0, 1]$ .

[F1]

Componentwise $L^{2}$ and its pairing. For $f = (f_{1}, \dots, f_{r})$ with components in the scalar space $L^{2} (D; C)$ of The space $L^{p} (μ)$ as the quotient by null functions and Complex Lp classes and Euclidean test-function conventions, the integral $⟨ f, g ⟩ = \sum_{l = 1}^{r} \int_{D} f_{l} \overline{g_{l}} d x$ is finite and defines an inner product, linear in the first argument, with $∥ f ∥^{2} = \sum_{l} ∥ f_{l} ∥^{2}$ ; for every measurable $B$ the partial pairing satisfies $∣ ⟨ f, g ⟩_{B} ∣ \leq ∥ f ∥_{L^{2} (B)} ∥ g ∥_{L^{2} (B)} \leq ∥ f ∥ ∥ g ∥$ ( $L^{2}$ with the integral pairing is a Hilbert space, Cauchy-Schwarz inequality for $L^{2}$ , Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors). Two choices of representatives of $f, g$ change $x \mapsto ⟨ f (x), g (x) ⟩_{C^{r}}$ and $x \mapsto ∥ f (x) + g (x) ∥^{2}$ only on a $d x$ -null set, so the pointwise assertions are well posed almost everywhere.

[F2]

Indefinite integrals are measures. For nonnegative measurable $h$ the set function $B \mapsto \int_{B} h d x$ is a measure on $(D, A)$ (The indefinite integral of a nonnegative measurable function is a measure); for $g \in L^{1} (d x)$ the real and imaginary parts of $B \mapsto \int_{B} g d x$ are countably additive by monotone convergence (Monotone convergence for the integral).

[F3]

Monotonicity of the integral. If $0 \leq h \leq H$ are measurable, then $\int_{D} h d x \leq \int_{D} H d x$ (Monotonicity and nonnegative homogeneity of the nonnegative integral).

[F4]

Probability measures and normalization. A probability measure is a measure on $(D, A)$ with total mass $1$ (Probability measures and probability spaces, Measures on sigma-algebras). A finite measure $ν \geq 0$ with $0 < ν (D) < \infty$ is normalized to a probability measure by $\overset{ˉ}{ν} (B) = ν (B) / ν (D)$ , because countable additivity divided by the constant $ν (D)$ stays countably additive and $\overset{ˉ}{ν} (D) = 1$ .

Proof

technique · direct; expand the squared component norm pointwise, then read off in turn the two normalized laws and the weight bounds, integrate the cross term, normalize it, and compute the coincidence condition from the two displayed formulas

Given: The data of the statement and [F1]-[F4].

1.1givenF1algebra

[Pointwise identity] Fix representatives of $a, b$ . For every $x \in D$ the identity $∥ a (x) + b (x) ∥_{C^{r}}^{2} = ∥ a (x) ∥_{C^{r}}^{2} + ∥ b (x) ∥_{C^{r}}^{2} + 2 Re ⟨ a (x), b (x) ⟩_{C^{r}}$ holds by the expansion of the finite-dimensional inner product; by [F1] the three real functions $x \mapsto ∥ a (x) ∥^{2}$ , $∥ b (x) ∥^{2}$ and $Re ⟨ a (x), b (x) ⟩$ are measurable, and the last is integrable because $∣ Re ⟨ a (x), b (x) ⟩ ∣ \leq ∥ a (x) ∥ ∥ b (x) ∥$ with $∥ a ∥ ∥ b ∥ \in L^{1} (d x)$ ; representatives differing on null sets change the identity only on a null set. This proves the pointwise assertion of (a).

1.2F2F4algebra

[Incoherent law] Assume $∥ a ∥, ∥ b ∥ > 0$ . By [F2] each of $B \mapsto \int_{B} ∥ a ∥^{2} d x$ and $B \mapsto \int_{B} ∥ b ∥^{2} d x$ is a finite measure with positive total masses $∥ a ∥^{2}$ and $∥ b ∥^{2}$ ; by [F4] each normalized set function is a probability measure, and a convex combination with $w \in [0, 1]$ of two probability measures is again a probability measure, because countable additivity is preserved and $w \cdot 1 + (1 - w) \cdot 1 = 1$ . This is (c).

1.3F2F3F4algebra

[Recorded weights] Applying [F3] with $h = W$ and $H = 1$ gives $ν (D) = \int_{D} W d μ \leq \int_{D} 1 d μ = 1$ and $ν (B) \geq 0$ for every $B$ because $W \geq 0$ ; countable additivity of $ν$ follows from [F2] applied to the measure $μ$ . If $ν (D) > 0$ , [F4] makes $\overset{ˉ}{ν}$ a probability measure. For the finite family, $ν_{a} = \int_{D} W_{a} d μ \geq 0$ and $\sum_{a} ν_{a} = \int_{D} \sum_{a} W_{a} d μ \leq \int_{D} 1 d μ = 1$ by finite additivity and [F3]; if $\sum_{a} W_{a} = 1$ pointwise, the middle integral equals $μ (D) = 1$ . This is (d).

2.1step 1.1F1algebra

[Integrated identity] Let $B \in A$ . Integrating the identity of step 1.1 over $B$ is legitimate because all three terms are integrable on $B$ by step 1.1 and [F1]; the integral of the left side is $\int_{B} ∥ a + b ∥^{2} d x$ and the integral of the right side is $\int_{B} ∥ a ∥^{2} d x + \int_{B} ∥ b ∥^{2} d x + 2 Re ⟨ a, b ⟩_{B}$ , which is the displayed identity of (a).

3.1step 2.1F2F3F4algebra

[Coherent law] Assume $∥ a + b ∥ > 0$ . The set function $B \mapsto \int_{B} ∥ a + b ∥^{2} d x$ is a finite measure with total mass $∥ a + b ∥^{2}$ by [F2], and $∥ a + b ∥^{2} \in (0, \infty)$ ; dividing by this constant gives a finite measure of total mass one, hence a probability measure by [F4], whose value at $B$ is $p_{c o h} (B)$ with denominator $N + 2 c$ by step 2.1 applied to $B = D$ . This is (b).

4.1step 1.2step 1.3step 2.1step 3.1F1F2algebra∎

[Coincidence criterion] Assume the hypotheses of (e), including $∥ a + b ∥ > 0$ . By steps 2.1 and 1.2 with $w = w_{*}$ , $p_{*} (B) = \frac{1}{N} \int_{B} (∥ a ∥^{2} + ∥ b ∥^{2}) d x$ and $p_{c o h} (B) = \frac{\int_{B} (∥ a ∥^{2} + ∥ b ∥^{2}) d x + 2 c (B)}{N + 2 c}$ with $c (B) : = Re ⟨ a, b ⟩_{B}$ ; since $N > 0$ and $N + 2 c = ∥ a + b ∥^{2} > 0$ , the equality $p_{c o h} (B) = p_{*} (B)$ is equivalent to $N c (B) = c M (B)$ for the finite measure $M (B) : = \int_{B} (∥ a ∥^{2} + ∥ b ∥^{2}) d x$ . The signed measure $N c (\cdot) - c M (\cdot)$ vanishes on every $B \in A$ if and only if its density $N Re ⟨ a (x), b (x) ⟩_{C^{r}} - c (∥ a (x) ∥^{2} + ∥ b (x) ∥^{2})$ vanishes $d x$ -almost everywhere by [F2], since a nonzero integrable density has a level set of strictly positive or strictly negative integral. This is the displayed criterion; if $⟨ a (x), b (x) ⟩ = 0$ a.e., then $c = 0$ and $c (B) = 0$ for every $B$ , so the laws coincide, and if $c = 0$ the criterion reduces to $Re ⟨ a (x), b (x) ⟩ = 0$ a.e. This proves (e), and steps 1.1, 1.2, 1.3, 2.1, 3.1 prove the remaining clauses.

Remarks

Convention. The pairing is linear in the first argument, as fixed by the cited $L^{2}$ items, so the cross term of a sum $a + b$ is $2 Re ⟨ a, b ⟩$ and not $2 Re ⟨ b, a ⟩$ ; the two agree because the real part is symmetric.
What is not asserted. The lemma is an identity and inequality package for classes and weights; it produces no detector law, no state update, no sample of trials and no measured frequency. The coincidence criterion concerns the two declared laws, not the question of which preparation a laboratory realizes.

Component amplitudes, coherent and incoherent bin laws, and recorded detector weights

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

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