Let \(\mu>0\) be an arbitrarily fixed real and \(B\) a \(d\)-dimensional Brownian motion defined on a probability space \((\Omega,{\mathcal F},P)\). By \({\mathcal P}\) we denote the set of all probabilities \[ Q= \exp \Biggl\{\int_0^T\theta_t\,dB_t- \tfrac12\cdot \int_0^T| \theta_t|^2\,dt \Biggr\}\cdot P, \] where \(\theta\) runs all \(({\mathcal F}^B_t)\)-adapted processes bounded by \(\mu\). The authors study the properties of the minimal expectation defined for \(\xi\in \bigcup_{p>1}L^p(\Omega,{\mathcal F}^B_T,P)\) by \({\mathcal E}[\xi]=\inf_{Q\in {\mathcal P}}E_Q[\xi]\) and also its associated minimal conditional expectation. Using a result of \textit{N. El Karoui, S. Peng} and \textit{M. C. Quenez} [Math. Finance 7, No. 1, 1--71 (1997; Zbl 0884.90035)], \({\mathcal E}[.]\) is identified with the \(g\)-expectation \({\mathcal E}_g[.]\) (for \(g(z)=-\mu| z|\)) introduced by \textit{S. Peng} [in: Backward stochastic differential equations. Pitman Res. Notes Math. Ser. 364, 141--159 (1997; Zbl 0892.60066)]. The properties of \({\mathcal E}\) are consequently those of \({\mathcal E}_g.\) Apart from these properties the authors study also the Brownian motion \(B\) w.r.t. \({\mathcal E}\). A lot of explicit examples are given to illustrate their results.