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I'm trying to figure out how to compute a particular integral using Lebesgue integration.

For a number $a$, define $f(x) = x^a$ for $0 \lt x \leq 1$, and $f(0) = 0$. Compute $\int_0^1 f$.

Here is what I have so far:
$\int_0^1 f = \int_{(0,1]}f + 0 = (x^a)*m((0,1]) = x^a$

I'm not sure if I'm doing this correct. I'd appreciate some help, thanks in advance. This problem appears in Real Analysis by Royden (4th Edition) on p. 84, Exercise 19.

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    Then $\int_0^1$f = $\infty$2011-03-16

1 Answers 1

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Your problem seems to a very basic misunderstanding of the difference between definite integrals and functions.

$\int_{(0,1]} f$ is a specific number, not a function.

You cannot just do $\int_{(0,1]} f = f*m((0,1])$.

The statement that $\int_{(0,1]} f = x^a * m((0,1])$ is meaningless.

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    thankyou for ur help. i was getting confused with the lebesgue integration.2013-04-29