# Definition of Projection Measure in the characterization of strong approximation Resistance in a paper by Khot et al

I'm reading a paper about Constraint Satisfaction Problems, specifically "A Characterization of Strong Approximation Resistance", Subhash Khot, Madhur Tulsiani, Pratik Worah (ECCC TR13-075).

The authors give a characterization of strong approximation resistance which I don't completely understand.

Given a measure $\Lambda$ in $R^{k + {k \choose 2}}$, (the coordinates of a point in this space are written $x_1, x_2, ..., x_k, x_{(1,2)}, ... x_{(i,j)}, ... , x_{(k-1, k)}$) and some subset $S \subset [k]$, they define $\Lambda_S$ as the measure projected onto the coordinates of $S$.

I can think of two such interpretations for $\Lambda_S$. First fix $k=4$ so that $\Lambda$ is in $R^{10}$. Our coordinates are $1, 2, 3, 4, (1,2), (1,3), (1,4), (2,3), (2,4), (3,4)$. Fix $S = \{1, 2, 3\}$.

Interpretation 1: Under one interpretation, the measure $\Lambda_S$ should be projected using the coordinates $1$ and $2$ and $3$ only.

Interpretation 2: Under the other interpretation we should also include the "pair" coordinates so we should project using $1$, $2$, $3$, $(1,2)$, $(1,3)$, $(2,3)$.

I'm thinking that the second interpretation is the correct one. This is because the definition is reformulated in terms of matrices in definition 2.11 on page 19. In this formulation, we have a measure $\zeta$ on a subspace of matrices in $R^{(k+1)\times(k+1)}$ analogous to the measure $\Lambda$, but given $S$, projection measure $\zeta_S$ is defined by taking the submatrix formed by restricting to rows and columns whose indices are in $S$. But in this formulation, the entry $i,j$ of the matrix corresponds to the coordinate $x_{(i,j)}$.

Am I correct? Or is there some way to reconcile the first interpretation with the matrix analogue?