14 questions linked to/from equality between two sigma algebras
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### Preimage of generated $\sigma$-algebra

For some collection of sets $A$, let $\sigma(A)$ denote the $\sigma$-algebra generated by $A$. Let $C$ be some collection of subsets of a set $Y$, and let $f$ be a function from some set $X$ to $Y$. ...
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### Induced Borel $\sigma$-algebra.

Suppose that $X$ is a topological space equipped with the Borel $\sigma$-algebra $\mathcal{B}_X$. Let $Y$ be a Borel subset of $X$, $Y\in \mathcal{B}_X$. In particular, $Y$ is a topological space ...
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### Proving if $X$ is a random variable and $f$ is a continuous function, then $f(X)$ is a random variable

I'm reading "Fundamental of Mathematical Statistics" by Gupta and Kapoor and the authors make the claim that "if $X$ is a random variable and $f$ is a continuous function, then $f(X)$ is a random ...
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