The introduction of game-based learning techniques has significantly swayed learning, motivation, and information processing in both traditional and digital learning environments. This paper studies the footprint that the implementation of ten short-duration digital escape rooms has had on the creation of an environment of positive emotions in the educational field. The digital escape rooms were created by employing the Genial.ly platform and RPG Maker MZ software. A feelings/satisfaction questionnaire has been conducted to study what emotions students have experienced, as well as the students’ opinions about essential elements of digital escape rooms, to study whether positive feelings predominate in the performance of these activities. Results show a high incidence of positive emotions, and a very favorable opinion on the tools employed and the positive feelings on the acquisition of knowledge and skills.

If R is a ring of subsets of a set ? and ba (R) is the Banach space of bounded finitely additive measures defined on R equipped with the supremum norm, a subfamily ? of R is called a Nikod?m set for ba (R) if each set {?? : ???} in ba (R) which is pointwise bounded on ? is norm-bounded in ba (R). If the whole ring R is a Nikod?m set, R is said to have property (N), which means that R satisfies the Nikod?m-Grothendieck boundedness theorem. In this paper we find a class of rings with property (N) that fail Grothendieck?s property (G) and prove that a ring R has property (G) if and only if the set of the evaluations on the sets of R is a so-called Rainwater set for ba(R). Recalling that R is called a (wN)-ring if each increasing web in R contains a strand consisting of Nikod?m sets, we also give a partial solution to a question raised by Valdivia by providing a class of rings without property (G) for which the relation (N)?(wN) holds.

We call a subset $\mathcal{M}$ of an algebra of sets $\mathcal{A}$ a Grothendieck set for the Banach space $ba\left(\mathcal{A}\right)$ of bounded finitely additive scalar-valued measures on $\mathcal{A}$ equipped with the variation norm if each sequence ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }$ in $ba\left(\mathcal{A}\right)$ which is pointwise convergent on $\mathcal{M}$ is weakly convergent in $ba\left(\mathcal{A}\right)$ , i.e., if there is $\mu \in ba\left(\mathcal{A}\right)$ such that ${\mu }_n\left(A\right)\to \mu \left(A\right)$ for every $A\in \mathcal{M}$ then ${\mu }_n\to \mu$ weakly in $ba\left(\mathcal{A}\right)$ . A subset $\mathcal{M}$ of an algebra of sets $\mathcal{A}$ is called a Nikodým set for $ba\left(\mathcal{A}\right)$ if each sequence ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }$ in $ba\left(\mathcal{A}\right)$ which is pointwise bounded on $\mathcal{M}$ is bounded in $ba\left(\mathcal{A}\right)$ . We prove that if $\mathsf{\Sigma }$ is a $\sigma$ -algebra of subsets of a set $\mathsf{\Omega }$ which is covered by an increasing sequence $\left\{{\mathsf{\Sigma }}_n:n\in \mathbb{N}\right\}$ of subsets of $\mathsf{\Sigma }$ there exists $p\in \mathbb{N}$ such that ${\mathsf{\Sigma }}_p$ is a Grothendieck set for $ba\left(\mathcal{A}\right)$ . This statement is the exact counterpart for Grothendieck sets of a classic result of Valdivia asserting that if a $\sigma$ -algebra $\mathsf{\Sigma }$ is covered by an increasing sequence $\left\{{\mathsf{\Sigma }}_n:n\in \mathbb{N}\right\}$ of subsets, there is $p\in \mathbb{N}$ such that ${\mathsf{\Sigma }}_p$ is a Nikodým set for $ba\left(\mathsf{\Sigma }\right)$ . This also refines the Grothendieck result stating that for each $\sigma$ -algebra $\mathsf{\Sigma }$ the Banach space ${\ell }_{\infty }\left(\mathsf{\Sigma }\right)$ is a Grothendieck space. Some applications to classic Banach space theory are given.

A local convex space E is said to be distinguished if its strong dual $E_{\beta }^{\prime }$ has the topology $\beta (E^{\prime },{(E_{\beta }^{\prime })}^{\prime })$, i.e., if $E_{\beta }^{\prime }$ is barrelled. The distinguished property of the local convex space $C_p\left(X\right)$ of real-valued functions on a Tychonoff space X, equipped with the pointwise topology on X, has recently aroused great interest among analysts and $C_p$-theorists, obtaining very interesting properties and nice characterizations. For instance, it has recently been obtained that a space $C_p\left(X\right)$ is distinguished if and only if any function $f\in {\mathbb{R}}^X$ belongs to the pointwise closure of a pointwise bounded set in $C\left(X\right)$. The extensively studied distinguished properties in the injective tensor products $C_p\left(X\right){\otimes }_{\epsilon }E$ and in $C_p(X,E)$ contrasts with the few distinguished properties of injective tensor products related to the dual space $L_p\left(X\right)$ of $C_p\left(X\right)$ endowed with the weak* topology, as well as to the weak* dual of $C_p(X,E)$. To partially fill this gap, some distinguished properties in the injective tensor product space $L_p\left(X\right){\otimes }_{\epsilon }E$ are presented and a characterization of the distinguished property of the weak* dual of $C_p(X,E)$ for wide classes of spaces X and E is provided.