dp–convergence and

Consider a sequence of Riemannian manifolds $(M^n_i,g_i)$ with scalar curvatures and entropies bounded below by small constants $R_i,\mu_i \geq-\epsilon_i$. The goal this paper is to understand notions convergence the structure limits for such spaces. Even in seemingly rigid case $\epsilon_i\to 0$, we construct examples showing that may converge wildly Gromov-Hausdorff or Intrinsic Flat sense. On other hand, will see these classical are incorrect ones consider. Indeed, even metric space wrong underlying category be working on. Instead, introduce $d_p$ convergence, weaker notion valid class rectifiable These spaces have well-behaved topology, measure theory, analysis, though potentially there no reasonably associated distance function. Under closeness, almost nonnegative curvature entropy bounds must fact close Euclidean space; constitute our $\epsilon$-regularity theorem. More generally, compactness theorem saying sequences lower \geq -\epsilon$ $X$. Comparing first paragraph, functions $M_i$ degenerating, well-defined sense analysis cannot be. Applications include an $L^\infty$-Sobolev embedding apriori $L^p$ $p<1$.