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I have a function $\rho(f,g)$ to be the metric function for any two measurable functions $f,g$.

What does it mean by $f_n\to f$ in a metric and $f_n\to f$ in measure, where $(f_n)$ is a sequence of functions.

Shannon
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    There is some missing context here that more than likely affects the interpretation of these words. Apparently $f,g$ are known to be two "measurable functions", but the Reader would like to know more about their domain. Presumably they are real-valued functions, from the tag "lebesgue-measure", but this too would bear explicit confirmation. Finally there are many function spaces where one might define a "metric" giving distances between two functions. Are we told how $\rho(f,g)$ is specifically defined? – hardmath Nov 13 '14 at 03:12

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$f_n \to f$ in the metric $\rho$ means that $\lim_{n \to \infty} \rho(f_n, f)=0$.

$f_n \to f$ in the measure $\mu$ means that for all $\varepsilon >0$ you have $\lim_{n \to \infty} \mu ( \{ x : |f_n(x) - f(x)| > \varepsilon\} ) = 0$

Crostul
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