Top exterior powers are one-dimensional. For composable maps \(T,S\),
use the same intermediate form to obtain
\((ST)^*\eta =d(S)d(T)\eta _V\). Thus \(J(ST)=J(S)J(T)\);
applying this to \(T^{-1}T\) gives \(J(T^{-1})=J(T)^{-1}\).
The absolute identities are independent of enumeration. Since \(T^{-1}\)
is continuous, \(T(E)\) is Borel.
On the Borel real line, translation preserves length and multiplication
by \(a\ne 0\) has pushforward coefficient \(|a|^{-1}\). Compare preimages
of bounded intervals and then use finite-on-bounded-interval uniqueness.
When \(a<0\), the reversed half-open interval has the same length since
singletons have measure zero. By rectangles and Lemma [fdip-002I],
scaling one coordinate has that coefficient. Coordinate permutations
preserve measure by Proposition [fdip-002J]. With a common basis their signed
top-form multipliers are \(a\) for scaling and \(-1\) for a swap.
For distinct coordinates \(i,j\), the shear
\(E(x)=x+c\,x(j)\delta _i\) preserves volume. Split off coordinate \(i\),
writing the others as \(z\). For fixed \(z\), translation invariance gives
\[\int _{\mathbb R}F(z,t+c\,z(j))\,dt
=\int _{\mathbb R}F(z,t)\,dt.\]
Nonnegative Tonelli on the sigma-finite uncompleted Borel product then
gives equality of full integrals; indicators give measure preservation.
The shear's top-form multiplier is \(1\), since replacing the \(j\)th
basis vector by \(\delta _j+c\delta _i\) leaves its top wedge unchanged.
Tonelli here is
[tao2011measure, Theorem 1.7.15, pp. 200-201, preliminary version],
not the complete-space variant.
These elementary maps generate every isomorphism in indicator charts.
Induct on \(n\), the zero-dimensional case being immediate. For \(n>0\),
the image of the first basis vector is nonzero. Swap target coordinates
to put a nonzero coefficient first, scale it to one, and subtract its
suitable multiples from the other target coordinates. The resulting
map \(ET\) sends the first domain basis vector to the first target one.
If the first coefficient of \((ET)e_j\) is \(a_j\) for \(j>1\), precompose
with the domain shears \(e_j\mapsto e_j-a_je_1\). The resulting map
fixes the first basis vector and maps the complementary span into the
target complementary span. Its restriction is injective; it is surjective
because the first coefficient of a preimage of a complementary vector
must be zero. Induction on these complements, extending their elementary
maps by identity on the first coordinate, completes the factorization.
Pushforwards compose and commute with scalar multiplication of
measures. Multiplying the elementary coefficients therefore proves
\(T_*\lambda _V=J(T)^{-1}\lambda _W\), after the measure-preserving
coordinate charts of Proposition [fdip-002J]. Substituting \(B=T(E)\) gives
the image formula. Translations in several coordinates follow from the
one-dimensional result and rectangles, yielding the affine assertion.
No reciprocal formula is asserted for a singular map.
The integral formula follows first for indicators from the measure
identity, then for nonnegative simple functions by additivity, then by
increasing simple approximation and monotone convergence. Apply it to
\(|F|\) to obtain absolute-integrability equivalence, since \(J(T)\) is
positive and finite. Only then apply it to \(F^+,F^-\) and subtract.
No multiplication closure of \(L^1\) is used.