Consider a Radon measure \(\mu\) on the line and a constant \(0 < \alpha < 1\). The \(\alpha\)-dimensional average densities of \(\mu\) are introduced by \textit{T. Bedford} and \textit{A. M. Fisher} [Proc. Lond. Math. Soc., III. Ser. 64, No. 1, 95-124 (1992; Zbl 0741.28004)]. More precisely, the lower and upper circular average densities of \(\mu\) at \(x \in {\mathbb R}\) are defined by \[ \underline{D}^{\alpha}(\mu, x) = \liminf_{\varepsilon \to 0} {1 \over | \log \varepsilon| } \int_{\varepsilon}^1 {{\mu([x-t, x+t])}\over {t^{\alpha}}}{dt \over t}, \] and \[ \overline{D}^{\alpha}(\mu, x) = \limsup_{\varepsilon \to 0}{1 \over | \log \varepsilon| } \int_{\varepsilon}^1 {\mu([x-t, x+t]) \over t^{\alpha}} {dt \over t}. \] The lower and upper left-sided average densities \(\underline{D}_{-}^{\alpha} (\mu, x)\) and \(\overline{D}_{-}^{\alpha} (\mu, x)\) are defined in the same way by replacing the symmetric interval \([x-t, x+t]\) by \([x-t, x]\), and the lower and upper right-sided average densities \(\underline{D}_{+}^{\alpha} (\mu, x)\) and \(\overline{D}_{+}^{\alpha} (\mu, x)\) are defined by replacing \([x-t, x+t]\) by \([x, x+t]\). If \(\underline{D}^{\alpha}(\mu, x) = \overline{D}^{\alpha}(\mu, x)\), then the common value is called the average density of \(\mu\) at \(x\) and is denoted by \({D}^{\alpha}(\mu, x)\). In the paper under review, the author proves that if \(\mu\) has positive lower and finite upper \(\alpha\)-densities \(\mu\)-almost everywhere, then the following relations hold \(\mu\)-almost everywhere: \[ \underline{D}_{-}^{\alpha} (\mu, x) = \underline{D}_{+}^{\alpha} (\mu, x) = (1/2)\cdot \underline{D}^{\alpha} (\mu, x) \] and \[ \overline{D}_{-}^{\alpha} (\mu, x) = \overline{D}_{+}^{\alpha} (\mu, x) = (1/2)\cdot \overline{D}^{\alpha} (\mu, x) \] These answer a question of Bedford and Fisher. The author derives these equalities from the following more general result on the tangent measure distributions of \(\mu\) about \(x\): for \(\mu\)-almost every \(x\), the formula \[ \iint G(\nu, u) d\nu(u) dP(\nu)= \iint G(T^{u}\nu, -u) d\nu(u) dP(\nu) \] holds for every tangent measure distribution \(P\) of \(\mu\) at \(x\) and all Borel functions \(G: {\mathcal M}({\mathbb R})\times {\mathbb R} \to [0, \infty)\). Here \(T^{u}\nu\) is the measure defined by \(T^{u}\nu(E) = \nu(u + E)\) and \({\mathcal M} ({\mathbb R})\) is the space of Radon measures with the vague topology. It is also proven that the tangent measure distributions are Palm distributions and thus define \(\alpha\)-self-similar random measures in the sense of \textit{U. Zähle} [Probab. Theory Relat. Fields 80, No. 1, 79-100 (1988; Zbl 0638.60064)].