Is there a relationship between the probabilistic interepretation of Sherali-Adams SDP hierarchy and the Lasserre SDP hierarchy?

• Firstly note this paper http://ttic.uchicago.edu/~madhurt/Papers/reductions.pdf where a Lasserre SDP is being setup for the independent set probblem at the bottom of page 4 where the author says says, "It can be shown that for any set S with |S| ≤ t, the vectors $$U_{S'}$$, S' ⊆ S induce a probability distribution over valid independent sets of the subgraph induced by S."

I guess that the author is claiming the following identity,for all subsets $$S$$ of size at most $$t$$, $$\sum_{S' \subseteq S, S' \text{independent in the subgraph induced by }S} ||U_S'||_2^2 = 1$$ Can someone clarify/prove that this above statement follows from whatever has been defined about these vectors $$U_{S'}$$ on page 4? (Is the author claiming that this equality above is a consequence of how he has defined the feasible set of $$U_{S'}$$ vectors in the top box of page $$5$$?)

• I guess these vectors $$U_{S'}$$ can be interpreted as what I would call "Lasserre vectors" (is there any standard term for these?) (By Lassere vectors" I mean the rows of the matrix $$V$$ which factorizes the degree $$d$$ pseudo-moment matrix $$M$$ as $$M = VV^T$$) So is the author claiming that there is such a probabilistic interpretation for any Lassere vector?

• For the Sherali-Adams SDP hierarchy let me follow the definition of $$SA_r$$ as in the box at the top of page $$5$$ here, http://dsteurer.org/paper/cspgaps.pdf. (...I haven't seen any book or other older paper define this notion! People seem to typically refer to a certain LP hierarchy as the Sherali-Adams hierarchy!...)

Is there any relationship between this local integral distribution $$\mu_S$$ defined in this paper and the probability distribution alluded to in the first paper? Are the SDP vectors $$v_{i,a}$$ used in the $$SA_R$$ definition to be also thought of as or are even remotely related to Lasserre vectors?