Assuming that $(\Omega ,\Sigma ,\mu )$ is a complete probability space and $X$ a Banach space, in this paper we investigate the problem of the $X$-inheritance of certain copies of $c_0$ or ${\ell }_{\infty }$ in the linear space of all [classes of] $X$-valued $\mu$-weakly measurable Pettis integrable functions equipped with the usual semivariation norm.

For $i=(1,2)$, let $X_i$ be completely regular Hausdorff spaces, $E_i$ quasi-complete locally convex spaces, $E=E_1\breve{\otimes }{E}_2$, the completion of the their injective tensor product, $C_b\left(X_i\right)$ the spaces of all bounded, scalar-valued continuous functions on $X_i$, and ${\mu }_i$ $E_i$-valued Baire measures on $X_i$. Under certain conditions we determine the existence of the $E$-valued product measure ${\mu }_1\otimes {\mu }_2$ and prove some properties of these measures.

Jachymski showed that the set $$\left\{(x,y)\in {𝐜}_0\times {𝐜}_0:{\left(\sum _{i=1}^n\alpha \left(i\right)x\left(i\right)y\left(i\right)\right)}_{n=1}^{\infty }\text{is}\text{bounded}\right\}$$ is either a meager subset of ${𝐜}_0\times {𝐜}_0$ or is equal to ${𝐜}_0\times {𝐜}_0$. In the paper we generalize this result by considering more general spaces than ${𝐜}_0$, namely ${𝐂}_0\left(X\right)$, the space of all continuous functions which vanish at infinity, and ${𝐂}_b\left(X\right)$, the space of all continuous bounded functions. Moreover, we replace the meagerness by $\sigma$-porosity.

If $(\Omega ,\Sigma ,\mu )$ is a finite measure space and $X$ a Banach space, in this note we show that $L_{w^{*}}^1(\mu ,X^{*})$, the Banach space of all classes of weak* equivalent $X^{*}$-valued weak* measurable functions $f$ defined on $\Omega$ such that $\parallel f\left(\omega \right)\parallel \le g\left(\omega \right)$ a.e. for some $g\in L_1\left(\mu \right)$ equipped with its usual norm, contains a copy of $c_0$ if and only if $X^{*}$ contains a copy of $c_0$.