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Exercise 18 on p.39 in Exercises 2B in "Measure, Integration & Real Analysis" by Sheldon Axler.

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#real-analysis #measure-theory #measurable-functions #borel-measures
26
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$\forall r>0,j=1,2,\ldots:\mu(\partial B(x_j,r))=0$ for a Borel probability measure $\mu$ over a metric space $X$ where $x_j$ are countably dense

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#real-analysis #measure-theory #separable-spaces #borel-measures
75
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Arbitrary union of measure $0$ open sets has measure $0$ in a separable metric space

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#real-analysis #general-topology #measure-theory #borel-measures #second-countable
54
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If $B$ is a Borel set s.t. $\mu(\partial B)=0$ and $\lim_n\mu_n(B)=\mu(B)$ then does $\lim_n\mu(\overline{B}^{1/n}\setminus\overline{B})=0$ hold?

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#real-analysis #limits #measure-theory #borel-measures
58
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Calculating a lower bound of Lebesgue measure of some Borel set

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#measure-theory #lebesgue-measure #diophantine-approximation #borel-measures
60
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A detail on the Polar Coordinate Theorem proof

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#measure-theory #lebesgue-measure #outer-measure #borel-measures
32
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Borel Measurability of a subset of $R^{\infty}$

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#borel-sets #borel-measures #measurable-sets
42
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regarding measurability of functions

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#measure-theory #lebesgue-measure #borel-measures
287
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Does a set of positive outer measure contain a *measurable* set of positive measure?

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#measure-theory #outer-measure #borel-measures
82
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How can I prove that if $\int_{[0,1]} f~d\nu\leq C\int_{[0,1]} f~d\mu$ then the measures are absolutely continuous?

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#functional-analysis #analysis #measure-theory #borel-measures #absolute-continuity
40
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Borel measure - unbounded sets

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#general-topology #borel-measures
134
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proove that $\forall x \in X : \forall \epsilon > 0 : \exists \delta > 0 \text{ s.t. } \mu(B(x,\delta)) ≤ \epsilon$

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#measure-theory #metric-spaces #borel-measures
50
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Is $\mu_U(E) := \mu(E\cap U)$ a Radon measure when $U$ is open?

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#real-analysis #measure-theory #borel-measures
170
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About the definition of a Borel measurable function in "Measure, Integration & Real Analysis" by Sheldon Axler.

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#real-analysis #measure-theory #soft-question #borel-measures
65
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If $f,g$ are two measurable functions on $[0,1]$ such that $\int_A f=\int_A g$ for all $A\subset[0,1]$ such that $\mu(A)=1/3$. Then $f=g$ a.e. $\mu$

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#real-analysis #analysis #measure-theory #lebesgue-measure #borel-measures

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