Theorem: The vector space $L^1$ is complete in its metric. The following proof is from Princeton Lectures in Analysis book $3$ page $70$. Some of my questions about the proof of this theorem are as follows. First assume a Cauchy sequence $(f_n)\in L^1$, then we try to extract a subsequence $\left(f_{n_k}\right)$ of $(f_n)$ which converges […]

A measure on $\mathbb R$ is a set function $\mu,$ defined for all Borel sets of $\mathbb R,$ which is countably additive(that is, $\mu(E)=\sum \mu(E_{i})$ if $E$ is the union of the countable family $\{E_{i}\}$ of pairwise disjoint Borel sets of $\mathbb R$), and for which $\mu(E)$ is finite if the closure of $E$ is […]

I was thinking if it is possible to come up with a $\sigma$-finite measure on $\mathbb R$ which is positive on any uncountable set. I think that I have a proof that there is no such measure – but I am not sure if it is formal enough. The proof: let $\mu$ be such measure, […]

Let $(f_n)$ be a sequence in $L_p[0,1]$ ($1<p<\infty$) such that $f_n\to f\in L_p[0,1]$ pointwise almost everywhere. If there exists $M>0$ such that $\|f_n\|_p\le M$ for all $n$, and $g\in L^q[0,1]$, where $\dfrac{1}{p}+\dfrac{1}{q}=1$, then: $$\int fg=\lim_{n\to\infty}\int f_ng.$$ I tried this with the Hölder’s inequality: $$\left|\int fg-\int f_ng\right|\le \|f_n-f\|_p\|g\|_q.$$ I’m not sure we can say directly that […]

Let $T\in B(H)$ is normal, and $B(\sigma(T))$ denote the $C^*$ algebra of all bounded Borel functions on $\sigma(T)$. Then is it true that $B(\sigma(T))$ is a closed $C^*$ algebra under the sup. norm ? Actually I try to prove the Spectral representation Theorem of a normal operator in case the representation of $C^*(T)$ over $H$ […]

Give an example of sequence of Measurable functions defined on some measurable subset $E$ of $\mathbb{R}$ such that $f_{n} \to f$ pointwise almost everywhere on $E$ but $$\int\limits_{E} f \ dm \not\leq \lim_{n} \inf \int\limits_{E} f_{n} \ dm$$

The monotone class theorem states that for any algebra of sets $\cal A$ one can construct the smallest monotone class generated by this class ${\cal M}(\cal A)$. This smallest monotone class is also the smallest sigma-algebra $\Sigma(\cal A)$ generated by $\cal A$, and ${\cal M}(\cal A)=\Sigma(\cal A)$. However, I noticed that in applications of the […]

Let me recall some notations: The mean oscillation of a locally integrable function $u$ (i.e. a function belonging to $ L^1_{\textrm{loc}}(\mathbb{R}^n))$ over a cube Q in $\mathbb R^n$ (which has sides parallel to axis) is defined as the following integral: $\frac{1}{|Q|}\int_{Q}|u(y)-u_Q|\,\mathrm{d}y$, where $|Q|$ is the volume of $Q$, i.e. its Lebesgue measure, $u_Q$ is the […]

Here (Section “Integration in respect to a complex measure”), they say that: Another approach is to not develop a theory of integration from scratch, but rather use the already available concept of integral of a real-valued function with respect to a non-negative measure. To that end, it is a quick check that the real and […]

Based on Williams’ Probability w/ Martingales: Let $(S, \Sigma, \mu)$ be a measure space. Dominated Convergence Theorem: Suppose $\{f_n\}_{n \in \mathbb{N}}$, $f$ are $\Sigma$-measurable $\forall n \in \mathbb{N}$ s.t. $\lim_{n \to \infty} f_n(s) = f(s) \forall s \in S$ or a.e. in S and $\exists g \in \mathscr{L}^1 (S, \Sigma, \mu)$ s.t. $|f_n(s)| \le g(s) […]

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