# OSA: Theory of Integration

## Question 1

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

claimcorrect?resultexplanation

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): $\lim_{n \to \infty} \int_{[0,n]} e^{-\sqrt{x^2+n^{-2}}}\,\mathrm{d}x$