  Compensated compactness is an important method used to solve nonlinear PDEs.
A simple formulation of a compensated compactness problem is to ask for
conditions on a set $\mathcal{K}\subset M^{m\times n}$ such that $$
\lim_{n\rightarrow \infty}
\mathrm{dist}(Du_n,\mathcal{K})\overset{L^p}{\rightarrow} 0\; \Rightarrow
\{Du_{n}\}_{n}\text{ is precompact.} $$ Let $M_1,M_2,\dots, M_q$ denote the set
of minors of $M^{m\times n}$. A sufficient condition for this is that any
measure $\mu$ supported on $\mathcal{K}$ satisfying $$ \int M_k(X) d\mu
(X)=M_k\left(\int X d\mu (X)\right)\text{ for }k=1,2,\dots, q $$ is a Dirac
measure. We call measures that satisfy the above equation "Null Lagrangian
Measures" and we denote the set of Null Lagrangian Measures supported on
$\mathcal{K}$ by $\mathcal{M}^{pc}(\mathcal{K})$. For general $m,n$, a
necessary and sufficient condition for triviality of
$\mathcal{M}^{pc}(\mathcal{K})$ was an open question even in the case where
$\mathcal{K}$ is a linear subspace of $M^{m\times n}$. We answer this question
and provide a necessary and sufficient condition for any linear subspace
$\mathcal{K}\subset M^{m\times n}$. The ideas also allow us to show that for
any $d\in \left\{1,2,3\right\}$, $d$-dimensional subspaces $\mathcal{K}\subset
M^{m\times n}$ support non-trivial Null Lagrangian Measures if and only if
$\mathcal{K}$ has Rank-$1$ connections. This is known to be false for $d\ge 4$.
  Using the ideas developed we are able to answer (up to first order) a
question of Kirchheim, M\"{u}ller and Sverak on the Null Lagrangian measures
arising in the study of a (one) entropy solution of a $2\times 2$ system of
conservation laws that arises in elasticity.
