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!

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