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I'd would like to confirm if the following proposition is indeed true in the case of an arbitrary measure space. Theorem: Let $(X,\Sigma,\mu)$ be a measure space and $\{f_n\}_{n\in\mathbb{N}}\subsete …

According to the 7.2.2 Theorem of the book "Measure Theory" written by V.I. Bogachev, every tight $\tau$-additive finite measure is Radon. The proof says: "The restrictions of a $\tau$-additive measur …

According to Theorem 1 of [1], the stastistic $(X_1,\cdots,X_n)$ is minimal sufficient for the statistical model $X_1,\cdots,X_n\sim N(\theta,1)$ iid and $\theta\in\mathbb{R}$. This is false, as you c …

According to the 7.3.5. Corollary of the book "Measure Theory" by V.I. Bogachev we have the following result: Let $(X,\tau)$ be a completely regular space and let $\Gamma$ be a family of continuous f …

I would like example of measures which shows that the following propositions are false: Proposition 1: Let $\mathfrak{B}$ be the Borel $\sigma$-algebra of a topological space $X$ and $\mu:\mathfrak{B …

Let $X,Y,Z$ be three standard measurable spaces and $f:X\to Z$ and $g:X\to Y$ two measurable functions. Suppose that there's a function $h:Y\to Z$ such that $f=h\circ g$. How can I show that there's a …

The Kuratowski Extension Theorem says: Let $(X,\mathcal{A})$ be a measurable space, $Y$ be a polish space, $A\subseteq X$, and $f:A\to Y$ be a measurable map. Then there is a measurable function $F:X\ …

Is there a measurable function $g:\mathbb{R}\to \mathbb{R}^n$ such that $g(\sum_{i=1}^nx_i)=(x_1,\dotsc,x_n)$ a.e.? Due to the papers [1], [2], and [3] I'm obtaining a result that I think it's false. …

I'd like to know if there's a proof of the Dunford-Pettis theorem without using relatively advanced theorems of functional analysis such as Eberlein–Smulian Theorem. Since I'm only interested in proba …

Could someone please indicate me some reference that contains the proof of the following theorem? Below $\mathcal{H}^n$ denotes the $n$-dimensional Hausdorff outer measure in $\mathbb{R}^n$. Theorem: …