• Federer density
• measure theoretic area formula

The present thesis deals with the Federer density, which is a specific function depending on a measure defined on a metric space and an another measure built throw a gauge function. Both measures are defined in the same metric space. Federer density was introduced for the first time in an article of Prof.Magnani, in order to find a measure theoretic area formula in metric spaces. The goal of the thesis is to present a complete discussion on Federer density and some applications. In particular in the thesis a new version of the theorem is elaborated. By adding some natural hypotheses, we have extended the measure theoretic area formula from the Borel case to the measurable case.