Covering theorems and integration (Q1772996)

Let \(\Omega\subset\mathbb R^d\) be a measurable set, \(\mu\) a Radon measure and \(S\) a \(\rho\)-regular Morse cover of \(\Omega\). In this paper, the following result is proved: For each \(\varepsilon > 0\), there exists a gauge \(\gamma\) such that whenever \(\{S_n\}\) is a sequence in \(S\) with \(S_n(x)\subset\gamma(x)\) and \(\mu(\Omega\setminus\bigcup S_n(x))=0\), we have \[ \left|\sum_n f(x_n)\mu(S_n)- \int_\Omega f\,d\mu\right| <\varepsilon. \]