How to derive $\int_{\Sigma}\delta(*F)\wedge\delta A =\int_{\Sigma}d\Sigma^{\mu}\delta F_{\mu\nu}\wedge\delta A^{\nu}$ for free EM? I am reading Andy's lectures.
On page 22, one has the symplectic form for free electrodynamics as
$$\Omega_{\Sigma}=-\frac{1}{e^2}\int_{\Sigma}\delta(*F)\wedge\delta A,\tag{1}$$
where $\Sigma$ is any Cauchy hypersurface (3-dimensional) and $\delta$ is the variation in the phase space. Subsequently, (1) gives the equation with indices as
$$\Omega_{\Sigma}=-\frac{1}{e^2}\int_{\Sigma}d\Sigma^{\mu}\delta F_{\mu\nu}\wedge\delta A^{\nu},\tag{2}$$
where $d\Sigma^{\mu}$ is the induced mearsure times the unit norm vector to $\Sigma$.
Anybody know how to derive Eq.(2) from Eq.(1)? Further, what does it mean by "induced measure" here? Does it mean $\sqrt{|h|}dx^1dx^2dx^3$, where $h_{\mu\nu}$ is the induced metric? If it is indeed the case, then when the surface is null hypersurce $\mathcal{I}^{\pm}$, we have $h=0$.
 A: Let us here only address the last conceptional part of OP's question (v4).
Let there be given a 4-dimensional orientable Lorentzian manifold $(M,\mathbb{g})$ with canonical  pseudovolume$^1$ form $$\Omega~:=~\sqrt{|g|}\mathrm{d}x^{0}\wedge \mathrm{d}x^{1}\wedge\mathrm{d}x^{2}\wedge\mathrm{d}x^{3}~\in~\Gamma(\bigwedge \! {}^4(T^{\ast}M)), \qquad g~:=~\det(g_{\mu\nu}). \tag{A}$$
[Note that this Omega has nothing to do with the symplectic form in eqs. (1) & (2)]
Hamiltonian quantization in field theory typically rely on a choice of a vector field $X\in \Gamma(TM)$ that represents the flow of an evolution parameter, cf. e.g. this Phys.SE post. For a time-like vector field we can always normalize it, if we so wish; but if it is light-like, there is no canonical choice of $X$.
We next pick a Cauchy surface $\Sigma\subset M$ of co-dimension 1, i.e. a hypersurface, such that 
$$T^{\ast}_p\Sigma~=~{\rm Ker}(X_p), \qquad p~\in~\Sigma.\tag{B}$$
Here we have identified the vector $$X_p: T^{\ast}_pM~\to~ \mathbb{R}\tag{C}$$ with a functional on the cotangent space $T^{\ast}_pM$. 
We can then define a pseudovolume form $\omega \in \Gamma(\bigwedge\! {}^3(T^{\ast}\Sigma))$ on the Cauchy surface $\Sigma$ via a contraction 
$$\omega_p~:=~i_{X_p}\Omega_p, \qquad p~\in~\Sigma.\tag{D}$$
--
$^1$ A pseudovolume form transforms as
$$ \Omega^{\prime} ~=~{\rm sgn }(J)~\Omega, \qquad J~:=~\det(\frac{\partial x^{\prime \nu}}{\partial x^{\mu}}), \tag{E} $$
under general coordinate transformations $x^{\mu} \to x^{\prime \nu}=f^{\nu}(x)$.
