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 $h=0$.