Integral of Integrable Function over Null Set

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Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $f: X \to \overline \R$ be a $\mu$-integrable function.

Let $N$ be a $\mu$-null set.


Then:

$\displaystyle \int_N f \rd \mu = 0$

where $\displaystyle \int_N$ signifies an integral over $N$.


Proof


Sources