Lebesgue Measure and Measure

Prove that the measures $\mu((0,1])\lambda,\mu$ are equal to the algebra of sets $\mathcal{A}=$ $\{$ finite unions of intervals $[a,b)$ such that $-\infty \leq a<b \leq +\infty$ $\}$ .

Then since this measures are sigma finite,the by a known theorem they are equal to $\sigma(\mathcal{A})=\mathcal{B}_{\Bbb{R}}$

Then use the fact that every Lebesgue measurable set is a union of a borel set and a set of Lebesgue measure zero.

Both measures are equal to sets of measure zero.

Tags:

Probability Theory

Measure Theory

Lebesgue Measure

Related

proving the identity $b^3 = 6\binom{b}{3} + 6\binom{b}{2} + b$ for $b \in \mathbb{N}$, $b > 2$, by counting a set in different ways Help with $\int \frac{\sqrt{x^2 + 1}}{x}\:dx$ Existence of a subgroup $H$ of a finite group $G$ with $|H|>\sqrt{|G|}$ Phi Function Summation and Divisors If $f(x)=ax+b$ is linear, why is $f(\gamma x) \neq \gamma f(x)$ Prove $\forall n\ge0,43\mid 6^{n+2}+7^{2n+1}$ in three ways Prove that for any number $m$ there is always $n$, such that $m - n$ and $m + n$ are prime numbers Is "tautology" a syntactical notion? What is $\lim_{n\to\infty} \root{2n+1} \of {-1} ?$ Find the the least integral value of a for which all the roots of the equation $x^4-4x^3 -8x^2 +a=0$ are real Can the group $\mathbb Z \times \mathbb Z$ be written as union of finitely many proper subgroups of it? When will a group be Abelian?

Recent Posts