Let f be a function defined on a set X with values in a set Y . Let FX be aσ-algebra of subsets of X. Show that the setFY = {E ⊆ Y | f −1(E) ∈ FX }is a σ-algebra of subsets of Y
Question
Let f be a function defined on a set X with values in a set Y . Let FX be aσ-algebra of subsets of X. Show that the setFY = {E ⊆ Y | f −1(E) ∈ FX }is a σ-algebra of subsets of Y
Solution
To show that FY is a σ-algebra of subsets of Y, we need to verify three properties:
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Y ∈ FY: Since f is a function defined on X with values in Y, the pre-image of Y under f, denoted by f^(-1)(Y), is X. Since FX is a σ-algebra of subsets of X, X ∈ FX. Therefore, Y ∈ FY.
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If E ∈ FY, then Y\E ∈ FY: If E ∈ FY, then f^(-1)(E) ∈ FX. Since FX is a σ-algebra, it follows that X\f^(-1)(E) ∈ FX. But X\f^(-1)(E) = f^(-1)(Y\E). Therefore, Y\E ∈ FY.
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If (Ei)i∈I is a countable collection of sets in FY, then ∪i∈IEi ∈ FY: If each Ei ∈ FY, then f^(-1)(Ei) ∈ FX for each i. Since FX is a σ-algebra, it follows that ∪i∈If^(-1)(Ei) ∈ FX. But ∪i∈If^(-1)(Ei) = f^(-1)(∪i∈IEi). Therefore, ∪i∈IEi ∈ FY.
Since FY satisfies all three properties, it is a σ-algebra of subsets of Y.
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