The set consisting of numbers in (0, 1) having a decimal expansion containing one hundred consecutive 4s is a Borel subset of R.
To show that the set A consisting of those numbers in the open interval (0, 1) that have a decimal expansion containing one hundred consecutive 4s is a Borel subset of R, we need to understand the concept of Borel sets and their properties.
A Borel set is any set that can be formed from open intervals using operations such as countable unions, countable intersections, and complements. The Borel σ-algebra on R is the smallest σ-algebra that contains all open intervals.
Let A be the set in question, defined as follows:
A = {x
(0, 1) | decimal expansion of x contains one hundred consecutive 4s}
We can express A as a countable union of open intervals.
Let
be the set of numbers in (0, 1) whose decimal expansion contains a sequence of n consecutive 4s. Then,
A =

Each
is an open set because, for each n, you can define an open interval around each such number that contains the sequence of n consecutive 4s. Since a countable union of open sets is a Borel set, A is a Borel set.
In conclusion, the set A is a Borel subset of R because it can be expressed as a countable union of open intervals, which generates a Borel set according to the definition of Borel sets.