2.6 MEASURE THEORY
Measure theory is concerned with assigning a “size” called a measure to subsets of a sample space Ω. Depending on the problem, this size might be a length, an area, or a volume in Euclidean space; this is known as the Lebesgue measure which is defined below. In our case, we are interested in assigning a probability measure to subsets of Ω that comprise the σ-field 
. For uncountable experiments, it is not possible to assign probabilities in a consistent manner to 
. The power set of 
is too large (it has cardinality beth two 
), so instead we choose 
to be 
, which is the Borel σ-field on the real line and has cardinality beth one 
(the same as 
). The Borel σ-field is useful for ...
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