Meaning of vanishes outside a set of finite measure

[guildmage]

meaning of "vanishes outside a set of finite measure"

What does it mean when we say that a function vanishes outside a set of finite measure? As in the definition of the integral as a prelude to the Lebesgue integral in Royden's Real Analysis, 3rd ed. (p. 77). It said:

If $$\phi$$ vanishes outside a set of finite measure, we define the integral of $$\phi$$ by

$$\int{\phi(x)dx} = \sum^{n}_{i=1}{{a_i}{mA_i}}$$​

 

[fourier jr]

it means $$\phi$$ is zero. think of the value of a characteristic function when x is outside the set of finite measure you're integrating over. hopefully i didn't embarrass myself this time :tongue2:

 

[LCKurtz]

It means there is a set E with finite measure such that f(x) = 0 if x is not in E.