  We relate $L^p$ convergence of metric tensors or volume convergence to a
given smooth metric to Intrinsic Flat and Gromov-Hausdorff convergence for
sequences of Riemannian manifolds. We present many examples of sequences of
conformal metrics which demonstrate that these notions of convergence do not
agree in general even when the sequence is conformal, $g_j=f_j^2g_0$, to a
fixed manifold. We then prove a theorem demonstrating that when sequences of
metric tensors on a fixed manifold $M$ are bounded, $(1-1/j)g_0 \le g_j \le K
g_0$, and either the volumes converge, $\operatorname{Vol}_j(M)\rightarrow
\operatorname{Vol}_0(M)$, or the metric tensors converge in the $L^p$ sense,
then the Riemannian manifolds $(M,g_j)$ converge in the measured
Gromov-Hausdorff and volume preserving Intrinsic Flat sense to $(M,g_0)$.
