The author studies the localisation problem for Bochner-Riesz means \(S^\alpha_R\) of order \(\alpha\ge 0\) which are defined as \(S_R^\alpha f(x):=\int_{|\xi|\le R} \Big(1- \frac{|\xi|^2}{R^2}\Big)_+^\alpha \hat f(\xi) e^{2\pi ix\xi} d\xi\) for suitable functions \(f\) in \(\mathbb R^d\). The fact that \(S^\alpha_R f(x)\) converges to zero uniformly on every compact subset of \(B\) for functions that vanish on an open bounded set \(B\), is referred to as the Riemann localisation principle. This is known to hold for \(d=1\) for functions \(f\in L^p(\mathbb R)\) for \(1\le p\le 2\). For \(d\ge 2\) it was shown by S. Bochner [Trans. Am. Math. Soc. 40, 175–207 (1936; Zbl 0015.15702)] that it may fail for every point when \(0\le \alpha< (d-1)/2\). A. Carbery and F. Soria [Rev. Mat. Iberoam. 4, No. 2, 319–337 (1988; Zbl 0692.42001)] proved that \(S_R^0f(x)\to 0\) almost everywhere on compact subsets of \(B\) and L. Colzani et al. [Trans. Am. Math. Soc. 366, No. 12, 6229–6245 (2014; Zbl 1309.42010)] studied the Hausdorff dimension of the sets of localisation of \(S_R^\alpha f\) for functions in \(L^2(\mathbb R^2)\) and \(0<\alpha<(d-1)/2\) depending on \(\alpha\). In the paper the author presents a different approach to study the Hausdorff dimension of the sets of localisation. His main result establishes that, under certain conditions on the parameters \(\alpha, \delta\) and \(d\), for a given \(\varepsilon>0\), there exists a constant \(c>0\) such that for every \(f\in L^2(\mathbb R^d)\) vanishing on an open set \(B\) and every nonnegative Borel measure \(d\mu\) with compact support \(K\subset B\) and \(d(K, \partial B)>\varepsilon\), one has \(\int_{\mathbb R^d} \sup_{R>0} |S^\alpha_R f(x)|d\mu (x)\le \|f\|_2(\int_{\mathbb R^d}|x-y|^{-\delta} d\mu(x)d\mu(y))^{1/2}.\) This result, due to Frostman’s lemma, which guarantees that for a given Borel set \(B\) of Hausdorff dimension \(\delta\), there exists a Borel measure \(d\mu\) supported on \(B\) such that \(\int_{\mathbb R^d}|x-y|^{-\delta} d\mu(x)d\mu(y)<\infty\), gives some restriction about the Hausdorff dimension of the set of points where \( |S^\alpha_R f(x)|\) can diverge. Also results in the same spirit are given for \(\mathbb T^d\) instead of \(\mathbb R^d\).