Probability density functions (PDFs) can be understood as continuous compositions by the theory of Bayes spaces. The origin of a Bayes space is determined by a given reference measure. This can be easily changed through the well-known chain rule which has an impact on the geometry of the Bayes space. This work provides a mathematical framework for setting a reference measure. It is used to develop a weighting scheme on the bounded domain of distributional data. The impact on statistical analysis is shown from the perspective of simplicial functional principal component analysis. Moreover, a novel centered log-ratio transformation is proposed to map a weighted Bayes spaces into an unweighted $L^2$ space, enabling to use most tools developed in functional data analysis (e.g. clustering, regression analysis, etc.) while accounting for the weighting strategy. The potential of our proposal is shown through simulation and on a real case study using Italian income data.