Sigma Algebra induced by an Indicator Function

Last updated Nov 1, 2022

Let $X$ be a Set. Let $A \in \mathcal{P}(X)$. Then $$\sigma(1_{A}) = {\emptyset, A, A^{C}, X}$$

We must check that $$\sigma(1_{A}) = {1_{A}^{-1}(E) : E \in \mathcal{B}(\mathbb{R}) } = {\emptyset, A, A^{C}, X}$$

Suppose $E \in \mathcal{B}(\mathbb{R})$. Then there are 4 possiblities

1. $E \cap {0, 1} = \emptyset$: Then $1_{A}^{-1}(E) = \emptyset$.
2. $E \cap {0, 1} = {0}$: Then $1_{A}^{-1}(E) = A^{C}$.
3. $E \cap {0, 1} = {1}$: Then $1_{A}^{-1}(E) = A$.
4. $E \cap {0, 1} = {0, 1}$: Then $1_{A}^{-1}(E) = X$.

Since these are the only 4 possibilities, we have that $$\sigma(1_{A}) = {1_{A}^{-1}(E) : E \in \mathcal{B}(\mathbb{R}) } = {\emptyset, A, A^{C}, X}$$ $\blacksquare$

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