Characteristic Function of Null Set is A.E. Equal to Zero/Corollary
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Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $N$ be a $\mu$-null set.
Then:
- $\chi_{X \setminus N} = 1$ $\mu$-almost everywhere.
where $\chi_{X \setminus N}$ is the characteristic function of $X \setminus N$.
Proof
From Characteristic Function of Set Difference, we have:
- $\chi_{X \setminus N} = \chi_X - \chi_{X \cap N}$
From Intersection with Subset is Subset, we therefore have:
- $\map {\chi_{X \setminus N} } x = 1 - \map {\chi_N} x$
for each $x \in X$.
From Characteristic Function of Null Set is A.E. Equal to Zero, we have:
- $\chi_N = 0$ $\mu$-almost everywhere.
So we have, from Pointwise Addition preserves A.E. Equality:
- $1 - \chi_N = 1$ $\mu$-almost everywhere.
So:
- $\chi_{X \setminus N} = 1$ $\mu$-almost everywhere.
$\blacksquare$