Final answer:
To prove that B is a sigma algebra over X, we must show that X is finite, since B contains sets that are finite or have finite complements. The definition of a sigma algebra requires that X, the universal set, be in B, which means X must be finite.
Step-by-step explanation:
The question asks to show that if B is a sigma algebra over X, and B is defined as B= A is finite or Aᵅ is finite, then the set X must be finite. To prove this, we use the properties of a sigma algebra. First, note that for any set A in our sigma algebra, its complement Aᵅ is also in B. Now, consider the set X itself, which must belong to B by the definition of a sigma algebra. Thus, either X is finite or Xᵅ is finite. However, since Xᵅ is the empty set, and the empty set is finite, this implies that X must also be finite. If X were infinite, its complement would not be finite, and hence, it would not satisfy the conditions required to be in B. Therefore, the assumption that B is a sigma algebra over X necessitates that X is finite