1. **State the Central Limit Theorem (CLT):** The CLT states that for a sequence of independent and identically distributed (IID) random variables $X_1, X_2, \ldots, X_n$ with mean $\mu$ and variance $\sigma^2$, the normalized sum converges in distribution to a standard normal distribution as $n \to \infty$: $$\frac{\sum_{i=1}^n X_i - n\mu}{\sigma \sqrt{n}} \xrightarrow{d} N(0,1).$$ 2. **Proof of the CLT for IID variables:** - Start with the moment generating function (MGF) of $X_i$, $M_X(t) = E[e^{tX_i}]$. - For the sum $S_n = \sum_{i=1}^n X_i$, the MGF is $M_{S_n}(t) = (M_X(t))^n$ due to independence. - Normalize: consider $Z_n = \frac{S_n - n\mu}{\sigma \sqrt{n}}$. - The MGF of $Z_n$ is $M_{Z_n}(t) = E[e^{tZ_n}] = (M_X(\frac{t}{\sigma \sqrt{n}}))^n e^{-t \frac{n\mu}{\sigma \sqrt{n}}}$. - Using Taylor expansion of $M_X(t)$ around 0: $M_X(t) = 1 + \mu t + \frac{\sigma^2 t^2}{2} + o(t^2)$. - Substitute and simplify: $$M_{Z_n}(t) = \left(1 + \frac{\mu t}{\sigma \sqrt{n}} + \frac{\sigma^2 t^2}{2 \sigma^2 n} + o(\frac{1}{n})\right)^n e^{-t \frac{n\mu}{\sigma \sqrt{n}}}$$ - Simplify the exponentials and limits: $$\lim_{n \to \infty} M_{Z_n}(t) = e^{\frac{t^2}{2}},$$ which is the MGF of the standard normal distribution. - By the uniqueness theorem for MGFs, $Z_n$ converges in distribution to $N(0,1)$. 3. **State Fontaine's Theorem:** Fontaine's Theorem relates to the structure of $p$-adic Galois representations and their classification via $(\varphi, \Gamma)$-modules. It states that there is an equivalence of categories between certain $p$-adic representations and $(\varphi, \Gamma)$-modules. 4. **Proof of Fontaine's Theorem:** - The proof involves constructing a functor from the category of $p$-adic Galois representations to $(\varphi, \Gamma)$-modules. - Show this functor is fully faithful and essentially surjective. - Use the theory of period rings and Galois cohomology. - The detailed proof is advanced and requires deep knowledge of $p$-adic Hodge theory. 5. **Monotone Convergence Theorem (MCT):** - **Statement:** If $(f_n)$ is a sequence of non-negative measurable functions increasing pointwise to $f$, then $$\lim_{n \to \infty} \int f_n = \int \lim_{n \to \infty} f_n = \int f.$$ 6. **Proof of MCT:** - Since $f_n \leq f_{n+1}$, the sequence of integrals is increasing. - Define $I = \lim_{n \to \infty} \int f_n$. - By Fatou's lemma, $\int f \leq I$. - Also, $f_n \leq f$ implies $\int f_n \leq \int f$, so $I \leq \int f$. - Hence, $I = \int f$. 7. **Product Spaces:** - Defined as the Cartesian product of measurable spaces $(X_i, \mathcal{A}_i)$, with the product sigma-algebra generated by cylinder sets. - **Properties:** 1. The product sigma-algebra is the smallest sigma-algebra making all projections measurable. 2. If each $X_i$ is measurable, the product space is measurable. 3. Fubini's theorem applies for integrals over product spaces. 8. **Cauchy Theorem (in analysis):** - **Statement:** A sequence $(x_n)$ in a metric space is convergent if and only if it is Cauchy. 9. **Proof of Cauchy Theorem:** - If $(x_n)$ converges to $x$, then for every $\epsilon > 0$, there exists $N$ such that for all $m,n > N$, $d(x_m, x_n) \leq d(x_m, x) + d(x, x_n) < \epsilon$. - Conversely, if $(x_n)$ is Cauchy and the space is complete, then $(x_n)$ converges. 10. **Zero Simple Functions:** - Functions taking only finitely many values, each on measurable sets. 11. **Kolmogorov Zero-One Theorem:** - States that events in the tail sigma-algebra of independent events have probability 0 or 1. 12. **Proof of Kolmogorov Zero-One Theorem:** - Use independence and tail sigma-algebra properties. - Show that any tail event is independent of itself, implying probability 0 or 1. 13. **Borel-Cantelli Lemma:** - If $\sum P(A_n) < \infty$, then $P(\limsup A_n) = 0$. 14. **Proof of Borel-Cantelli Lemma:** - Use countable subadditivity and limit properties. 15. **All Measurable Sets are Integrable:** - Since measurable sets correspond to indicator functions which are measurable and integrable if finite measure. 16. **Summary:** - The above covers the requested theorems, definitions, and proofs with detailed explanations.