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In measure theory, in the little wood three priciple it is noted that $f_n$ is nearly uniformly convergent. What is the meaning of this nearly. Also I would like to know

$A$ is nearly equal to $B$ means what?

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It is a vague word to help inspire intuition, without precise meaning in general. When applying any of the principles you are actually applying some precise theorem. In this case the precise meaning is the statement of Egorov's theorem.

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    $A$ is nearly equal to $B$ means what – 2017-02-14
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    @Seena: In what context? The answer will be similar, once you say what $A$ and $B$ are, if true the statement will be an imprecise version of some theorem to inspire intuition. E.g., measurable $f$ is nearly equal to a continuous function: See Lusin's theorem. Measurable set $E$ is nearly a finite union of intervals: I don't know if it has a name, but the precise statement is that for bounded measurable $E\subset \mathbb R$, for all $\varepsilon>0$ there is a finite union of intervals $U$ such that the symmetric difference of $E$ and $U$ has measure less than $\varepsilon$. – 2017-02-14
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    (I should have said $E$ is measurable with finite measure; boundedness isn't needed.) – 2017-02-14
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    Sir thanks for your reply. My question is for $E$ nearly a finite union of intervals why we take the Symmetric difference of $E$ and $U$ has measure less than $\epsilon$ – 2017-02-16
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    @SeenaV: Because having small symmetric difference is one measure of being "near". Because a finite union of intervals is easier to work with than a general measurable set. If we were allowed an arbitrary open set we could make $m(U\setminus E)<\varepsilon$, but that isn't possible with a finite union of intervals, and the symmetric difference being small is the next best thing. – 2017-02-16