Characteristic Function of Null Set is A.E. Equal to Zero/Corollary

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.