Asymptotics is strange...

It is a useful trick to “flip” the denominator into numerator when it comes to proving asymptotic properties of errors. Here’s how to do so. Consider a form
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[Real Analysis] Ch 2. Construction of Lebesgue Measure

In this chapter, we construct the Lebesgue measure on $\mathbb{R}^d.$ For this, we prove Riesz representation theorem and use the result to construct a complete measure space $(\mathbb{R}, \mathfrak{M}, m)$ such that integration with respect to $m$ is equal to Riemann integration for all Riemann-integrable functions. We then use $\sigma$-compactness of $\mathbb{R}$ to show that such $m$ is the Lebesgure measure.
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