Quartz sync: May 27, 2025, 12:31 PM

content/Exam Preparation.md

@@ -16,8 +16,8 @@ Here are 20 relevant exam questions in the topology and measure theory part of t
 9. Why does a real valued continuous function obtain its maximum on a compact set? [[Question 9|Answer]]
 10. What is a net? Given an example of an upward filtered ordered set. [[Question 10|Answer]]
 11. What is the initial topology? [[Question 11|Answer]]
-12. What is the product topology? [[Question 12|Answer]] - TODO
-13. What is a measure? Easy examples? [[Question 13|Answer]] - TODO: Example
+12. What is the product topology? [[Question 12|Answer]]
+13. What is a measure? Easy examples? [[Question 13|Answer]]
 14. Define the Lebesgue integral of a extended non-negative measurable function.
 15. State Lebesgue's monotone convergence theorem.
 16. Define $L^p$-spaces, and point out their crucial property.

content/Revision/Complex Analysis/Question 6.md

@@ -1,3 +1,4 @@
 # Question
 Taylor and Laurent series.
 # Answer
+## Taylor Series

content/Revision/Complex Analysis/Question 9.md

@@ -0,0 +1,5 @@
+# Question
+Applications to the computation of integrals.
+# Answer
+- Fourier Transform
+- 

content/Revision/Real Analysis/Question 12.md

@@ -1,3 +1,7 @@
 # Question
 What is the product topology?
 # Answer
+Let $X_{\lambda}$ be the [[Topological Space|topological spaces]].
+The [[Product Topology|product topology]] is on $\Pi X_{\lambda}$ is the [[Initial Topology|initial topology]] induced by the family of projections $\pi_{\lambda}$.
+
+$$\Pi_{\lambda \in I} X_{\lambda} \equiv \{ f:  \}$$

content/Revision/Real Analysis/Question 13.md

@@ -13,3 +13,4 @@ Then we say that $X$ is a measure space.
 > [!note]
 > You may also want to take a look at [[Measurable|measurable]]
 ## Easy Examples
+[[Lecture 14#Lemma]]

content/Revision/Real Analysis/Question 14.md

@@ -1,3 +1,13 @@
 # Question
 Define the Lebesgue integral of a extended non-negative measurable function.
 # Answer
+The [[Lebesgue Integral|Lebesgue integral]], $A \in M$ of a [[Measurable|measurable]] function $f : X \to [0, \infty]$ is
+$$\int_{A} f \, d\mu \equiv \sup_{0\leq s\leq f} \int_{A} s \, d\mu \in [0, \infty]$$
+> [!info] What is $s$?
+> $s$ is [[Measurable|measurable]] and a [[Simple Function|simple function]]
+> 
+> That is the simple function brings:
+> $s : X \to \mathbb{R}$ of the form $s = \sum_{i=1}^{n} a_{i} \times X_{a_{i}}$ for pairwise disjoint $A_{i} \subset X$
+
+
+

content/Revision/Real Analysis/Question 15.md

@@ -0,0 +1,6 @@
+# Question
+State Lebesgue's monotone convergence theorem.
+# Answer
+Say $X$ has a [[Measure|measure]] $\mu$, let $f_{n} : X \to [0, \infty]$ be [[Measurable|measurable]], and $f_{1} \leq f_{2} \leq f_{3} \leq \dots$ Then
+$$\int f_{m} \, d\mu \to \int \lim_{ n \to \infty } f_{n} \, d\mu$$
+as $m \to \infty$