1. ## Measure Theory

Any help with any of these would be very much appreciated, even if it's just a link to some useful material, etc. Thanks in advance!

1. Let $(S,\Sigma,\mu)$ be a $\sigma$-finite measure space. Suppose that $g \colon S \to [0,\infty]$ is a $\Sigma$-measurable function such that $\int f \cdot g \; d \mu \leq \int f \; d \mu$ for every integrable function $f \colon S \to [0,\infty)$. Prove that $g \leq 1$ $\mu$-almost everywhere on $S$.
2. Let $(S,\Sigma,\mu)$ be a $\sigma$-finite measure space. Let $f \colon S \to (0,\infty)$ be a $\Sigma$-measurable function with $\int f \; d \mu = 1$ and let $0 < r < 1$. Prove that for every $E \in \Sigma$ with $0 < \mu(E) < \infty$, $\int f^r d \mu \leq \int \mu(E)^{1-r}$.
3. Let $(S,\Sigma,\mu)$ be a $\sigma$-finite measure space. Let $g \colon S \to \mathbb{R}$ be integrable. Prove that there exists a bounded $\Sigma$-measurable function $f \colon S \to \mathbb{R}$ such that $\lVert f \rVert_\infty = 1$ and $\int f \cdot g \; d \mu = \lVert g \rVert_1$.
4. If $f \in L^{\infty}$, prove that $|f| \leq \lVert f \rVert_\infty$ $\mu$-almost everywhere. Moreover, prove that if $\alpha < \lVert f \rVert_\infty$, then there exists an $E \in \Sigma$ with $\mu(E) > 0$ and $|f(s)| \geq \alpha$ for all $s \in E$.
5. Prove that $(L^\infty (S,\Sigma,\mu), \lVert \cdot \rVert_\infty)$ is a normed linear space.

2. ## Homework Help??

I did not realize that Miami University graduate students in the department of mathematics turned to the internet for homework help 2 days before it is due.