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We prove that there exists an example of a metrizable non-discrete space $X$, such that $C_p(X\times \omega ){\approx }_l{C}_p\left(X\right)$ but $C_p(X\times S)\neg {\approx }_l{C}_p\left(X\right)$ where $S=(\left\{0\right\}\cup \{\frac{1}{n+1}:n\in \omega \})$ and $C_p\left(X\right)$ is the space of all continuous functions from $X$ into reals equipped with the topology of pointwise convergence. It answers a question of Arhangel’skii ([2, Problem 4]).

We prove: 1) Every Baire measure on the Kojman-Shelah Dowker space admits a Borel extension. 2) If the continuum is not real-valued-measurable then every Baire measure on M. E. Rudin's Dowker space admits a Borel extension. Consequently, Balogh's space remains the only candidate to be a ZFC counterexample to the measure extension problem of the three presently known ZFC Dowker spaces.

Let X and Y be two Polish spaces. Functions f,g: X → Y are called equivalent if there exists a bijection φ from X onto itself such that g∘φ = f. Using a theorem of J. Saint Raymond we characterize functions equivalent to Borel measurable ones. This characterization answers a question asked by M. Morayne and C. Ryll-Nardzewski.