I cannot understand the sentence in a probability book.

When Ω is uncountable (e.g., Ω = R or [0, 1]),
it is not possible to define a reasonable measure for every subset of
Ω; for
example, it is not possible to find a measure on all subsets of R and
still
satisfy property m([a, b]) = b - a. This is why it is necessary to
introduce σ-fields that
are smaller than the power set.

what is the meaning of the sentence?
Thanks
.

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