1. If $m(E) < \infty$, prove that
   $$\|f\|_{L^\infty} = \lim_{p\to\infty} \|f\|_{L^p}.$$

2. If $m(E) < \infty$, prove the inclusion
   $$L^p(E) \subset L^q(E), \quad \forall \, 0 < q \leq p \leq \infty.$$

3. Prove that for any $1 \leq p \neq q \leq \infty$, the inclusion does not hold: $L^p(\mathbb{R}^d) \not\subset L^q(\mathbb{R}^d)$.

4. For any $f \in L^p(E)$ and $0 < r < p < s \leq \infty$, show that there exists a decomposition $f = g + h$ such that $g \in L^r(E)$ and $h \in L^s(E)$.

5. For any $p, q, r \in [1, \infty]$ satisfying
   $$\frac{1}{p} = \frac{1}{q} + \frac{1}{r},$$
   prove that
   $$\|fg\|_{L^p} \leq \|f\|_{L^q} \|g\|_{L^r}.$$

6. Suppose a function $f : [0, \infty) \to \mathbb{R}$ satisfies, for constants $1 \leq p < \infty, \, 0 < q < \infty$, the condition
   $$\int_0^\infty |f(x)|^p x^{p-q-1} dx < \infty.$$
   Define $F(x) = \int_0^x f(t) dt$, and prove the inequality
   $$\int_0^\infty |F(x)|^p x^{p-q-1} dx \leq \left(\frac{p}{q}\right)^p \int_0^\infty |f(x)|^p x^{p-q-1} dx.$$

7. If $0 < q \leq p \leq s \leq \infty$, then there exists $\theta \in [0, 1]$ such that
   $$\frac{1}{p} = \frac{\theta}{q} + \frac{1-\theta}{s}.$$
   For any $f \in L^q \cap L^s$, we have $f \in L^p$ and
   $$\|f\|_{L^p} \leq \|f\|_{L^q}^\theta \|f\|_{L^s}^{1-\theta}.$$

8. Prove that the translation operator $\tau_h$ is continuous in $L^p$, i.e., prove that
   $$\lim_{h \to 0} \|\tau_h f - f\|_{L^p} = 0.$$

9. Suppose $f \in L^p([0, 1])$, $1 \leq p \leq \infty$, satisfies
   $$\int_0^1 t^k f(t) dt = 0, \quad \forall k = 0, 1, 2, \dots,$$
   prove that $f = 0$ almost everywhere.

10. Prove that $L^\infty$ is not separable.