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guildmage
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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 [tex]\phi[/tex] vanishes outside a set of finite measure, we define the integral of [tex]\phi[/tex] by
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 [tex]\phi[/tex] vanishes outside a set of finite measure, we define the integral of [tex]\phi[/tex] by
[tex]\int{\phi(x)dx} = \sum^{n}_{i=1}{{a_i}{mA_i}}[/tex]