Probability Theory: A Comprehensive Course

Achim Klenke

Chapter 4 The Integral - all with Video Answers

Educators

Section 1

Construction and Simple Properties

01:21

Problem 1

Now we do not assume $\mu(\Omega)<\infty$. Assume there exists an $a>0$ such that for any $A \in \mathcal{A}$ either $\mu(A)=0$ or $\mu(A) \geq a$. Show that the reverse inclusion to Theorem $4.19$ holds,
$$
\mathcal{L}^{p^{\prime}}(\mu) \subset \mathcal{L}^{p}(\mu) \quad \text { if } 1 \leq p^{\prime} \leq p \leq \infty.
$$

Hunza Gilgit

Numerade Educator

Problem 2

Let $1 \leq p^{\prime}<p \leq \infty$ and let $\mu$ be $\sigma$-finite but not finite. Show that $\mathcal{L}^{p}(\mu) \backslash \mathcal{L}^{p^{\prime}}(\mu) \neq \emptyset .$

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