List questions from borel-sets

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For what spaces does every countable Borel cover have a finite subcover?

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Is every Borel subset of a measurable set measurable?

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Lebesgue Measurable Set which is not a union of a Borel set and a subset of a null meager set?

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Is the projection of a Jordan measurable set Jordan measurable?

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Lebesgue Measure equal to any translation invariant Measure on the Borel sigma Algebra

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Proving existence of dense subset $D \subseteq \mathbb R$, so that $\{f \geq \alpha\} \in \mathcal{A}$, $\forall \alpha \in D$

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Borel sets. Need to prove.

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$U$ is open then find a sequence of closed sets $ I_{n+1} \subset I_{n}$

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Simple function formula

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Show $\mathcal{B}( \mathbb R)$ contains all countable subsets of $\mathbb R$

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Tips to show that the Random Variable $X$ is $\mathcal{B}(\mathbb R)-\mathcal{B}(\mathbb R)-$measurable

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Proving Borel set

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Is the function $f(x) = \lfloor x\rfloor$ is measurable?

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How to define the Lebesgue measure on the real line

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List Question

On Borel subsets of a topological space

For what spaces does every countable Borel cover have a finite subcover?

Is every Borel subset of a measurable set measurable?

Lebesgue Measurable Set which is not a union of a Borel set and a subset of a null meager set?

Is the projection of a Jordan measurable set Jordan measurable?

Lebesgue Measure equal to any translation invariant Measure on the Borel sigma Algebra

Proving existence of dense subset $D \subseteq \mathbb R$, so that $\{f \geq \alpha\} \in \mathcal{A}$, $\forall \alpha \in D$

Borel sets. Need to prove.

$U$ is open then find a sequence of closed sets $ I_{n+1} \subset I_{n}$

Simple function formula

Show $\mathcal{B}( \mathbb R)$ contains all countable subsets of $\mathbb R$

Tips to show that the Random Variable $X$ is $\mathcal{B}(\mathbb R)-\mathcal{B}(\mathbb R)-$measurable

Proving Borel set

Is the function $f(x) = \lfloor x\rfloor$ is measurable?

How to define the Lebesgue measure on the real line

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