Rudin's Real and Complex Analysis - up to Section 1.13 review:
Theorem 1.10 & Borel sets:
Theorem 1.10 establishes a smallest $\sigma$-algebra for any space $X$ formed by any collection of subsets of $X$. The proof is direct and simple. Motivation of smallest sigma algebra becomes clear in next section.
Borel sets are defined in a slightly convoluted way. Simpler definition is in WIKI for Borel sets. $F_\sigma$ and $G_\delta$ sets are defined in terms of countable union of closed sets and countable intersection of open sets. Gives a nice history of this nomenclature.
Every continuous mapping of $X$ is Borel measurable - aka Borel functions.
All these dovetail nicely into Theorem $1.12$
Set up is a space $X$ and a topological space $Y$ and a map $f$ between $X$ into $Y$. Then any collection of sets in $Y$ whose inverse maps belongs to sigma algebra, is a sigma algebra in $Y$. This gives a great way to establish sigma algebras in $Y$.
If $f$ is measurable, any Borel set $E \in Y$ with $f^{-1}(E)$, then $f^{-1}(E) \in \sigma \in X$.
Most applicable proposition (proposition c in the book) is if $Y=[-\infty,\infty]$ and $f^{-1}[(\alpha,\infty)]$ belongs to sigma algebra in $X$, then $f$ is measurable.
Proofs follow from the definition of measurable spaces.
Theorem 1.10 & Borel sets:
Theorem 1.10 establishes a smallest $\sigma$-algebra for any space $X$ formed by any collection of subsets of $X$. The proof is direct and simple. Motivation of smallest sigma algebra becomes clear in next section.
Borel sets are defined in a slightly convoluted way. Simpler definition is in WIKI for Borel sets. $F_\sigma$ and $G_\delta$ sets are defined in terms of countable union of closed sets and countable intersection of open sets. Gives a nice history of this nomenclature.
Every continuous mapping of $X$ is Borel measurable - aka Borel functions.
All these dovetail nicely into Theorem $1.12$
Set up is a space $X$ and a topological space $Y$ and a map $f$ between $X$ into $Y$. Then any collection of sets in $Y$ whose inverse maps belongs to sigma algebra, is a sigma algebra in $Y$. This gives a great way to establish sigma algebras in $Y$.
If $f$ is measurable, any Borel set $E \in Y$ with $f^{-1}(E)$, then $f^{-1}(E) \in \sigma \in X$.
Most applicable proposition (proposition c in the book) is if $Y=[-\infty,\infty]$ and $f^{-1}[(\alpha,\infty)]$ belongs to sigma algebra in $X$, then $f$ is measurable.
Proofs follow from the definition of measurable spaces.
You're putting the rest of us to shame! Glad to see you back to blogging about math!
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