= Solution
The squared invariant <Probability amplitude>[amplitude] weights each accessible final configuration. The final <On shell>[on-shell] <momentum> measure counts the available states, and the four-dimensional <Dirac delta function>[delta function] imposes <energy> and <Conservation of momentum>[momentum conservation]:
$$
d\Phi_n=(2\pi)^4\delta^{(4)}\left(P-\sum_jp_j\right)\prod_j\frac{d^3p_j}{(2\pi)^3\,2E_j}.
$$
To prove its <Lorentz invariance> for proper orthochronous transformations, write
$$
\boxed{\frac{d^3p}{2E_{\mathbf p}}=d^4p\,\delta(p^2-m^2)\Theta(p^0).}
$$
The four-volume measure has unit <Lorentz transformation>[Lorentz Jacobian], $p^2$ is invariant, and the future <mass shell> is preserved. Likewise $\delta^{(4)}(\Lambda Q)=\delta^{(4)}(Q)$ because $|\det\Lambda|=1$. The product therefore proves that $d\Phi_n$ is invariant. <Spin> averages and sums belong in the <Probability amplitude>[amplitude] as appropriate; identical final particles require a <symmetry> divisor if the integration covers a fully labelled <Lorentz-invariant phase space>[phase space].
The initial $1/(2E_i)$ factors come from relativistic <normalization> of the incoming states. They are not individually invariant, so the displayed coordinate-time <transition rate> must not be called a <Lorentz scalar> merely because its final <Lorentz-invariant phase space>[phase space] is invariant. A one-particle decay has proper width $\Gamma_0=(2M_0)^{-1}\int|\mathcal M|^2d\Phi_n$ in its rest frame, while its laboratory rate is $\Gamma_{\mathrm{lab}}=(M_0/E_0)\Gamma_0$. A scattering <cross section> instead uses the invariant incident flux $4\sqrt{(p_1\cdot p_2)^2-m_1^2m_2^2}$. This distinguishes invariance of <Lorentz-invariant phase space>[phase space] from transformation of a measured rate.
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