The analysis and measurement of poverty is a crucial and unsolved issue in the field of social science. This work aims to measure poverty as a multidimensional notion using a new composite indicator. However, subjective choices as different weighting schemes on the indicator's construction could affect their interpretation and policy. It is necessary to consider the possible weighting configurations randomly to overcome this problem, and it is proposed in this work as interval-based composite indicators based on the results. This work aims to obtain robust and reliable measures based on a relevant conceptual model of poverty we have identified, considering various factors as weightings. Methodologically speaking, it is proposed an original procedure for measuring poverty in which it is computed a different composite indicator for each simulated weighting scheme of the identified factors. The weighting scheme in the Monte-Carlo simulation randomly creates an interval-based composite indicator based on the results. The different intervals are compared using different criteria (upper bound, center, and lower bound), and various rankings help analyze extreme scenarios and policy hypotheses. Critical situations are identified in Sicilia, Calabria, Campania and Puglia. The results demonstrate a relevant and consistent indicator measurement and the shadow sector's relevant impact on the final measures.