Given a Borel probability measure \(\mu\) on \({\mathbb R}^n\), the core of multifractal analysis consists of computing the (Hausdorff) dimensional multifractal spectrum of \(\mu\), that is, \[ f_{\mu}(\alpha)=\text{ dim}\left\{x: \alpha_{\mu}(x):= \lim_{r\rightarrow 0}{\log\mu B(x,r)\over \log r}=\alpha\right\}, \] and then establishing whether the function \(f_\mu(\alpha)\) equals the Legendre transform \(\tau^{*}(\alpha)\) of a dimensional index \(\tau(q)\) (\(L^q\)-spectrum of \(\mu\)) associated with the \(q\)-th moments of the measure \(\mu\) (this is the so-called multifractal formalism). A common device has been employed in the literature to prove that the multifractal formalism holds for various types of measures, namely, constructing a continuum of \textit{auxiliary measures} \(\mu_q\), \(q\in {\mathbb R}\), satisfying \[ \lim_{r\rightarrow 0}{\log \mu B(x,r)\over \log r}=\alpha(q)=-\tau'(q),\;\mu_q\text{-a.e.}, \tag{i} \] and \[ \lim_{r\rightarrow 0}{\log \mu_q B(x,r)\over \log r}=\tau^*(\alpha(q))=q\alpha(q)+\tau(q),\;\mu_q\text{-a.e.} \tag{ii} \] In this paper rigorous versions of facts i) and ii) above are obtained for arbitrary Borel probability measures on \({\mathbb R}^n\) within the multifractal framework introduced by the author [\textit{L. Olsen}, Adv. Math. 116, No. 1, 82-196 (1995; Zbl 0841.28012)]. In general, identities (i) and (ii) need to be replaced formally by inequalities involving \(\liminf\) and \(\limsup\) and lateral derivatives. In particular, for \(q=1\) the paper's version of (i) refines previous results by \textit{Y. Hertaux} [Ann. Inst. H. Poincaré, Probab. Stat. 34, No. 3, 309-338 (1998; Zbl 0903.28005)] and \textit{S.-M. Ngai} [Proc. Am. Math. Soc. 125, No. 10, 2943-2951 (1997; Zbl 0886.28006)]. Under the extra assumption that the upper \(L^q\) spectrum \(\overline{\tau}\) has derivative at \(q=1\), fact (i) is proved to hold, which in turn implies that the entropy dimension of \(\mu\) coincides with \(-\bar{\tau}'(1)\). This result gives a general answer to a conjecture in the literature that has been proved in some particular cases so far. As an intermediary step, density theorems are derived , which are multifractal analogues of those obtained in [\textit{S. J. Taylor} and \textit{C. Tricot}, Trans. Am. Math. Soc. 288, 679-699 (1985; Zbl 0537.28003)] and in [\textit{X. Saint Raymond} and \textit{C. Tricot}, Math. Proc. Camb. Philos. Soc. 103, No. 1, 133-145 (1988; Zbl 0639.28005)] and may have interest in their own.