The Reshetnyak Continuity Theorem 10.3 was first proved in [226], whereas our argument is from [15]. The paper [94] contains more on different notions of convergence for measures.

Often, tangent measures are only considered as defined on the unit ball instead of the whole of the space (see, for example, Section 2.7 in [15]). This is, however, sometimes restrictive. Here, we use Preiss’s original theory as developed in [224], also see Chapter 14 of [183]. The original definition, however, explicitly excluded the zero-measure from \({{\mathrm{Tan}}}(\mu , x_0)\), which here we include. Proposition 10.5 is a slight improvement of Preiss’ existence theorem for non-zero tangent measures, see Theorem 2.5 in [224] or the appendix of [227]. The proof of Lemma 10.4 is adapted from Theorem 2.44 in [15]. Lemma 10.6 is originally due to Larsen, see Lemma 5.1 in [174].

We remark that the study of local properties of a measure via its tangent measures has its limits. Most strikingly, Preiss constructed a purely singular positive measure on a bounded interval (in particular a \(\mathrm {BV}\)-derivative) such that each of its tangent measures is a multiple of Lebesgue measure, see Example 5.9 (1) in [224]. Also see [219] for a measure that has *every* local measure as a tangent measure at almost every point.

The notion of area-strict convergence in \(\mathrm {BV}\) seems to be somewhat less well known than it deserves. However, as shown in the next chapter, it is the right one when considering integral functionals.

Alberti’s original proof [4] of what is now called Alberti’s rank-one theorem is via a “decomposition technique” together with the \(\mathrm {BV}\)-coarea formula; a streamlined version of his proof can be found in [90]. Another proof in two dimensions was announced in [5]. There is also now a nice short geometric proof [181]. Our argument is more in the spirit of PDE theory and has several other implications, see [92]. For more on the wave cone we refer to [98, 209, 210, 229, 267, 268].

The Kirchheim–Kristensen Theorem 10.13 was already announced in 2011 [161] with a simpler proof in a special case. The full proof appeared in [162]. The theorem actually holds in more generality, see Problem 10.10.