Probability assignment in a continuous space

           Suppose the sample space S is continuous and un-countable. Such a sample space arises when the outcomes of an experiment are numbers. For example, such sample space occurs when the experiment consists in measuring the voltage, the current or the resistance. In such a case, the sigma algebra consists of the Borel sets on the real line.

           Suppose and is a non-negative integrable function such that,

                                                      

           For any Borel set ,
           defines the probability on the Borel sigma-algebra B .
           We can similarly define probability on the continuous space of etc.