# Tagged Questions

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### General condition that Riemann and Lebesgue integrals are the same

I'd like to know that when Riemann integral and Lebesgue integral are the same in general. I know that a bounded Riemann integrable function on a closed interval is Lebesgue integrable and two ...
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### Show $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0$

Need to prove $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0$ and $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{y}{1+y^2}e^{-ay} dy =0$ Can ...
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### Can the theory of Lebesgue integration be extended in a way analogous to extending Riemann integrals to improper Riemann integrals?

I recently (last night) learned the definition of Lebesgue integration and one of the limitations I was told was that some improper Riemann integrals aren't Lebesgue integrable. It occurred to me ...
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### improper Riemann integral and Lebesgue integral

Let $f$ be a continuous function on $(0,1]$ and is defined as $f: [0,1] \to \mathbb R$. Show that if $f$ is lebesgue integrable on $[0,1]$, the improper Riemann integral \$\lim_{\epsilon \to 0} ...