my confusion comes with observation thatHe is using Gyongy's lemma (or was it theorem), also known as Markovian projection in finance, I believe first derived in a finance context by Dupire and/or Derman.$$ E \frac{1}{T} \int_0^T \sigma^2_t dt\,=\, \int_0^{\infty} E( \frac{1}{T} \int_0^T \sigma^2_t dt\, | S = K ) p ( K ) d K $$The upper half of the second page with 'intuition' looks suspiciously. The formula
where p ( K ) is the spot density at maturity T ... . should be more accurately explained. It looks like assumed that $$ \sigma^2_t = \sigma^2_t ( S ( t ) )$$
and "p ( K ) is the spot density at maturity T" should be interpreted that p ( T ) is the density of the S ( T ) . On the other hand
$$ \frac{1}{T} \int_0^T \sigma^2_t ( S ( t )) dt\,$$ depends on S ( * ) on [ 0 , T ]. Though It might be other interpretation that did not clear represented in the slides.
$$\sigma^2_t\,=\, \sigma^2_t ( S_t ) $$
i.e [$] \int_0^T \sigma^2_t [$] depends on distribution of the S ( t ) ] for all t on [ 0 , T ] and not [$]S_T[$] as it was stated. Of course one can assume that we talk about rv [$]\frac{1}{T} \int_0^T \sigma^2_t dt[$] which is definitely defined at the moment T. Nevertheless in this case condition in right hand side should be this random variable and not S = K as it was written