OSA: Theory of Integration


Question 1

Decide which of the following claims are correct. Note: By \(\lambda_d\) we denote the Lebesgue-measure on \(\mathbb{R}^d\).


If \(\mathcal{A}\) is a \(\sigma\)-algebra on a set \(X\), and \(A,B \subseteq X\) with \(A,B \not\in \mathcal{A}\), then \(A \cup B \not\in \mathcal{A}\).

Let \(\mathcal{A}\) be the powerset of \(\mathbb{N}\) (the set of natural numbers) and \[\mu(A) := \begin{cases}1&\text{if }A\text{ is infinite}\\0&\text{if }A\text{ is finite}\end{cases}\] Then \(\mu\) is an outer measure on \(\mathbb{N}\).

Let \(A \subseteq \mathbb{R}^2\) be compact for some \(d \in \mathbb{N}\), and \(f : \mathbb{R}^d \to \mathbb{R}^d\) continuous. Then \(\lambda_d\big(f(A)\big) < \infty\).

Every measure is an outer measure.

Let \(f:\mathbb{R}^2 \to \mathbb{R}^2\) be measurable. Then \(\int_{\mathbb{R}^2} f(x,y)\,\mathrm{d}\lambda_2(x,y) = \int_{\mathbb{R}}\int_{\mathbb{R}} f(x,y)\,\mathrm{d}\lambda_1(x)\mathrm{d}\lambda_1(y)\).

Question 2

Compute the following limit (integration is with respect to Lebesgue-measure): \[ \lim_{n \to \infty} \int_{[0,n]} e^{-\sqrt{x^2+n^{-2}}}\,\mathrm{d}x \]