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onthetopo
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What is the reason that 1/x is not lebesgue integrable where as 1/x^2 is integrable. You can use any theorems: monotone convergence, dominated convergence you want.
adriank said:Well, 1/x2 ≥ 1/x on (0, 1].
vigvig said:Even better, in the extended real number set, 1/x2 ≥ 1/x on [0, 1].
Ok let me clarify then. Take any x in (0,1].onthetopo said:I think the crucial step in your proof is 1/x >= phi(x)
where phi(x)=((a+e)^-1)X_A
May I ask why? This is a myth to me.
both statements with " = " or ">= "are equivalent statements. Remember from logic theory that (True OR False) is equivalent to True.onthetopo said:Thanks for the reply. But on the extended real set, 1/x=1/x^2 at x=0 right. both are equal to +inf
The function 1/x is not Lebesgue integrable because it does not satisfy the Lebesgue integrability criterion. This criterion states that for a function to be Lebesgue integrable, its integral must be finite. However, the integral of 1/x from 0 to 1 is infinite, making it not Lebesgue integrable.
The Lebesgue integrability criterion is a condition that a function must satisfy in order to be considered Lebesgue integrable. It states that the integral of the function over a given interval must be finite for the function to be Lebesgue integrable.
No, 1/x is not Riemann integrable either. This is because the Riemann integrability criterion also requires the integral of a function to be finite, and as mentioned before, the integral of 1/x from 0 to 1 is infinite.
The main difference between Lebesgue integrability and Riemann integrability is the criterion that each uses to determine if a function is integrable. While the Lebesgue integrability criterion only requires the integral to be finite, the Riemann integrability criterion also requires the function to be bounded. This means that a function can be Lebesgue integrable but not Riemann integrable.
Yes, there are other reasons why a function may not be Lebesgue integrable. For example, a function may not be Lebesgue integrable if it has an infinite number of discontinuities or if it oscillates too rapidly. These conditions can also cause the integral of a function to be infinite, making it not Lebesgue integrable.