The goal of this article is to prove Egoroff's Theorem [13]. However, there are not enough theorems related to sequence of measurable functions in Mizar Mathematical Library. So we proved many theorems about them. At the end of this article, we showed Egoroff's theorem.MML identifier: MESFUNC8, version: 7.8.10 4.100.1011

We discuss boundedness and compactness properties of the embedding $M_{\Lambda }^1\subset L^1\left(\mu \right)$, where $M_{\Lambda }^1$ is the closed linear span of the monomials $x^{{\lambda }_n}$ in $L^1\left([0,1]\right)$ and $\mu$ is a finite positive Borel measure on the interval $[0,1]$. In particular, we introduce a class of “sublinear” measures and provide a rather complete solution of the embedding problem for the class of quasilacunary sequences $\Lambda$. Finally, we show how one can recapture some of Al Alam’s results on boundedness and the essential norm of weighted composition operators from $M_{\Lambda }^1$...

We show that for "most" compact nonmetrizable spaces, the unit ball of the Banach space C(K) contains an uncountable 2-equilateral set. We also give examples of compact nonmetrizable spaces K such that the minimum cardinality of a maximal equilateral set in C(K) is countable.

One-sided versions of maximal functions for suitable defined distributions are considered. Weighted norm equivalences of these maximal functions for weights in the Sawyer's Aq+ classes are obtained.

The main object of this work is to describe such weight functions w(t) that for all elements $f\in L_{p,\Omega }$ the estimate $\parallel wf{\parallel }_p\ge K\left(\Omega \right)\parallel f{\parallel }_p$ is valid with a constant K(Ω), which does not depend on f and it grows to infinity when the domain Ω shrinks, i.e. deforms into a lower dimensional convex set ${\Omega }_{\infty }$. In one-dimensional case means that $K\left(\sigma \right):=K\left({\Omega }_{\sigma }\right)\to \infty$ as σ → 0. It should be noted that in the framework of the signal transmission problem such estimates describe a signal’s behavior under the influence of detection and amplification. This work...

Let $\varphi$ and $\psi$ be holomorphic self-maps of the unit disk, and denote by $C_{\varphi }$, $C_{\psi }$ the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition operators $C_{\varphi }-C_{\psi }$ from Bloch spaces to Bloch spaces in the unit disk. Compactness of the difference is also characterized.

We complete the different cases remaining in the estimation of the essential norm of a weighted composition operator acting between the Hardy spaces $H^p$ and $H^q$ for 1 ≤ p,q ≤ ∞. In particular we give some estimates for the cases 1 = p ≤ q ≤ ∞ and 1 ≤ q < p ≤ ∞.

We estimate the essential norm of a weighted composition operator relative to the class of Dunford-Pettis operators or the class of weakly compact operators, on the space ${\mathscr{H}}^{\infty }$ of Dirichlet series. As particular cases, we obtain the precise value of the generalized essential norm of a composition operator and of a multiplication operator.

Cet article est consacré à l’étude d’un problème lié au critère de Beurling Nyman sur l’hypothèse de Riemann. On y étudie la continuité de la projection de la fonction indicatrice de l’intervalle $]0,1]$ sur un sous-espace vectoriel variable de l’ensemble des fonctions dont le carré est intégrable sur la demi-droite réelle, engendré par des fonctions dilatées de la fonction partie fractionnaire. Plus généralement, $y$ étant un élément fixé d’un espace de Hilbert $H$, on étudie l’application qui à un convexe...

It is proved that every operator from a weak*-closed subspace of ${\ell }_1$ into a space C(K) of continuous functions on a compact Hausdorff space K can be extended to an operator from ${\ell }_1$ to C(K).

We study extension operators between spaces of continuous functions on the spaces $\sigma ₙ\left(2^X\right)$ of subsets of X of cardinality at most n. As an application, we show that if $B_H$ is the unit ball of a nonseparable Hilbert space H equipped with the weak topology, then, for any 0 < λ < μ, there is no extension operator $T:C\left(\lambda B_H\right)\to C\left(\mu B_H\right)$.