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We prove the result on Baire classification of mappings $f:X\times Y\to Z$ which are continuous with respect to the first variable and belongs to a Baire class with respect to the second one, where $X$ is a $PP$-space, $Y$ is a topological space and $Z$ is a strongly $\sigma$-metrizable space with additional properties. We show that for any topological space $X$, special equiconnected space $Z$ and a mapping $g:X\to Z$ of the $(n-1)$-th Baire class there exists a strongly separately continuous mapping $f:X^n\to Z$ with the diagonal $g$. For wide classes of spaces...

We use a new technique of measures on Boolean algebras to investigate narrow operators on vector lattices. First we prove that, under mild assumptions, every finite rank operator is strictly narrow (before it was known that such operators are narrow). Then we show that every order continuous operator from an atomless vector lattice to a purely atomic one is order narrow. This explains in what sense the vector lattice structure of an atomless vector lattice given by an unconditional basis is far...