Sphere integrals in angle coordinates

[The angular measure $\tau$] For Borel $E\subseteq S^2$ put $\tau (E):={\int }_{(0,\pi )\times (0,2\pi )}{\mathbf{1}}_E(\mathrm{\Phi }(\theta ,\phi ))\sin \theta d(\theta ,\phi )$, the integral existing in $[0,\mathrm{\infty }]$ because the integrand is nonnegative and measurable: $\mathrm{\Phi }$ is continuous, hence Borel measurable, and ${\mathbf{1}}_E\circ \mathrm{\Phi }$ is measurable by [F7]. The set function $\tau$ is countably additive by the monotone convergence theorem applied to the partial sums of ${\sum }_k{\mathbf{1}}_{E_k}$ for pairwise disjoint Borel $E_k$, and $\tau (S^2)={\int }_0^{\pi }{\int }_0^{2\pi }\sin \theta d\phi d\theta =2\pi \cdot 2=4\pi$ by [F5] and [F6]; hence $\tau$ is a finite Borel measure.

[$T$ is a bijection onto the slit space] $\mathrm{\Omega }:={\mathbb{R}}^3\setminus K=\{y\mathrm{\ne }0\}\cup \{y=0,x<0\}$ is open. The image of $T$ is contained in $\mathrm{\Omega }$: for $\phi \in (0,\pi )$ the second coordinate $r\sin \theta \sin \phi$ is positive, for $\phi \in (\pi ,2\pi )$ it is negative, and for $\phi =\pi$ one has $y=0$ with first coordinate $r\sin \theta \cos \pi =-r\sin \theta <0$. Conversely, let $x\in \mathrm{\Omega }$ and put $r:=\mathrm{\mid }x\mathrm{\mid }>0$ and $u:=x\mathrm{/}r\in S^2$. Since $u\mathrm{\notin }K$, the case $u=(0,0,\pm 1)$ is excluded (it has $y=0$, $x$-coordinate $0\ge 0$), so $\mathrm{\mid }{u}_3\mathrm{\mid }<1$; by [F6] there is a unique $\theta \in (0,\pi )$ with $\cos \theta =u_3$, and then $\sin \theta >0$ and $v:=(u_1,u_2)\mathrm{/}\sin \theta$ satisfies $v_1^2+v_2^2=1$ by Pythagoras [F6]. By the parametrisation of the unit circle [F6] there is a unique $\phi \in [0,2\pi )$ with $v=(\cos \phi ,\sin \phi )$; $\phi =0$ would give $u=(\sin \theta ,0,u_3)$ with $y=0$ and $x$-coordinate $\sin \theta >0$, i.e. $u\in K$, which is excluded, so $\phi \in (0,2\pi )$. Then $x=T(r,\theta ,\phi )$ with $(r,\theta ,\phi )$ uniquely determined, so $T$ is a bijection of $(0,\mathrm{\infty })\times (0,\pi )\times (0,2\pi )$ onto $\mathrm{\Omega }$.

[$T$ is $C^1$ with positive Jacobian] The components of $T$ are products and composites of the differentiable functions $\sin ,\cos$ and the coordinate functions by [F6], so $T$ is $C^1$ with total derivative the matrix of partial derivatives (The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder, Total differentiability gives a local $O(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ increment bound and therefore continuity), namely the columns ${\mathrm{\partial }}_{r}T=(\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta )$, ${\mathrm{\partial }}_{\theta }T=(r\cos \theta \cos \phi ,r\cos \theta \sin \phi ,-r\sin \theta )$ and ${\mathrm{\partial }}_{\phi }T=(-r\sin \theta \sin \phi ,r\sin \theta \cos \phi ,0)$; expanding the determinant along the third column and using ${\cos }^2\phi +{\sin }^2\phi =1$ and ${\sin }^2\theta +{\cos }^2\theta =1$ [F6] gives $\det DT=r^2\sin \theta ({\cos }^2\theta +{\sin }^2\theta )=r^2\sin \theta >0$ (The Jacobian determinant of a square-dimensional $C^1$ map is the determinant of its Jacobian matrix).

[$T$ is a $C^1$ diffeomorphism onto $\mathrm{\Omega }$] At each point of the domain $DT$ is invertible by step 1.3, so by the inverse function theorem [F3] $T$ restricts to a $C^1$ bijection of a neighbourhood of that point onto an open subset of $\mathrm{\Omega }$ with $C^1$ inverse. Since $T$ is injective with image $\mathrm{\Omega }$ by step 1.2, these local inverses agree on overlaps with the unique set-theoretic inverse $T^{-1}:\mathrm{\Omega }\to (0,\mathrm{\infty })\times (0,\pi )\times (0,2\pi )$, which is therefore $C^1$: near any point of $\mathrm{\Omega }$ it coincides with a local $C^1$ inverse. Hence $T$ is a $C^1$ diffeomorphism of open sets, and [F2] applies to it.

[The polar formula for $\tau$] Let $f:{\mathbb{R}}^3\to [0,\mathrm{\infty }]$ be Borel. By [F4], ${\lambda }_3(K)=0$ and ${\int }_{{\mathbb{R}}^3}f={\int }_{\mathrm{\Omega }}f$. Applying the change of variables formula [F2] to the diffeomorphism $T$ of step 2.1 and the nonnegative Borel function $f{\mathrm{\mid }}_{\mathrm{\Omega }}$ gives ${\int }_{\mathrm{\Omega }}f={\int }_{(0,\mathrm{\infty })\times (0,\pi )\times (0,2\pi )}f(T(r,\theta ,\phi ))r^2\sin \theta d(r,\theta ,\phi )$. By Tonelli's theorem [F5] the triple integral factors as ${\int }_0^{\mathrm{\infty }}{r}^2({\int }_{(0,\pi )\times (0,2\pi )}f(r\mathrm{\Phi }(\theta ,\phi ))\sin \theta d(\theta ,\phi ))dr={\int }_0^{\mathrm{\infty }}{r}^2{\int }_{S^2}f(r\omega )d\tau (\omega )dr$: for each fixed $r$ the function $\omega \mapsto f(r\omega )$ is nonnegative and Borel, and its $\tau$-integral equals the rectangle integral of $f(r\mathrm{\Phi }(\cdot ))\sin (\cdot )$ by the defining formula for $\tau$, extended from indicators to nonnegative Borel integrands by the monotone convergence theorem [F7]. Hence $\tau$ satisfies the polar formula of [F1].

[Identification and consequences] By the uniqueness clause of [F1] the measure satisfying the polar formula is $\sigma$; hence $\tau =\sigma$, which is exactly the identity of (a) for indicators and, by the monotone convergence theorem, for all nonnegative Borel $F$ [F7]; clause (b) is the computation $\tau (S^2)=4\pi$ of step 1.1. For (c), if $F\in L^1(S^2,\sigma ;\mathbb{C})$ then applying (a) to $\mathrm{\mid }F\mathrm{\mid }$ shows $F\circ \mathrm{\Phi }$ is integrable against $\sin \theta d(\theta ,\phi )$, because the integral of $\mathrm{\mid }F\mathrm{\mid }$ over $S^2$ is finite; splitting $F$ into real and imaginary parts and each into positive and negative parts, and applying (a) to the four nonnegative functions (or Fubini, [F5]) gives the identity for $F$.