Quartz sync: May 27, 2025, 9:02 AM

content/Revision/Real Analysis/Question 11.md

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-# Question
+# Question
+What is the initial topology?
+# Answer
+Given a set $X$ and an indexed family $(Y_{i})_{i \in I}$ of [[Topological Space|topological spaces]] with functions
+$$f_{i} : X \to Y_{i},$$
+the initial topology $\tau$ on $X$ is the [[Weakest Topology|weakest topology]] on $X$ such that
+$$f_{i} : (X, \tau) \to Y_{i}$$
+is [[Continuous|continuous]].
+
+> [!info] Indexed Family
+> An indexed family here means like an index in programming, such as 0, 1, 2, 3.
+> For example here $Y_{1}, Y_{2}, Y_{3}, \dots$
+

content/Revision/Real Analysis/Question 12.md

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+# Question
+What is the product topology?
+# Answer

content/Revision/Real Analysis/Question 13.md

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+# Question
+What is a measure? Easy examples?
+# Answer
+It is pretty much self explanatory with the name for what it is supposed to do.
+
+But a [[Measure|measure]] on a set $X$ (*maybe not required:* with $\Sigma$ as a [[Sigma-Algebra|sigma-algebra]] over $X$)is a function $\mu : M \to [0,\infty]$, such that:
+1. **Non-negativity:** for all $E \in \Sigma,\ \mu(E) \geq 0$. (*maybe not required*)
+2. $\mu(\emptyset) = 0$
+3. **Countable additivity:** $\mu(\cup_{n=1}^{\infty} A_{n}) = \sum_{n=1}^{\infty}\mu(A_{n})$ (for pairwise disjoint $A_{n} \in M$)
+
+Then we say that $X$ is a measure space.
+
+> [!note]
+> You may also want to take a look at [[Measurable|measurable]]
+## Easy Examples

content/Revision/Real Analysis/Question 14.md

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+# Question
+Define the Lebesgue integral of a extended non-negative measurable function.
+# Answer