Understanding Equivalence Relations & O, o, ~ Notation

[Gear.0]

Sorry for such a basic question, but I don't know what they mean by the O, o, and ~ in a book I am reading. I'll write out the whole thing to show what I am asking about as well as to give context.
Those symbols appear in ii) and iii) below. Also note that I wrote them here as having a subscript 'b', but in the book they are actually directly below the O's or the ~, I just couldn't figure you how to get the 'b' below them in LaTeX.

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PROPOSITION 3.5 (Comparison theorem) Let $g:\left[ a,b\right[\rightarrow\mathbb{C}$ be an integrable function, and let $f:\left[ a,b\right[\rightarrow\mathbb{C}$ be a measurable function. Then

i) if $\left| f \right| \leq \left| g \right|$, then $f$ is integrable on $\left[ a,b \right[$;
ii) if $f=O_{b}\left( g\right)$ or if $f=o_{b}\left( g\right)$ and if $f$ is integrable on any interval $\left[ a,c\right]$ with $c<b$, then $f$ is integrable on $\left[ a,b\right[$;
iii) if $g$ is non-negative, $f$~$_{b} g$, and $f$ is integrable on any interval $\left[ a,c\right]$ with $c<b$, then $f$ is integrable on $\left[ a,b\right[$.

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Basically I just have no clue what they mean by the $O_{b}$ and the little o as well, also the ~$_{b}$ is foreign to me.. I was able to find some info on the latter however, I think the '~' refers to an "equivalence relation".

BTW: for the O, o, and ~, the subscript b is actually displayed directly beneath the O, o, or ~ where it appears. I just could not seem to get it to go below in LaTeX.

 

[mathman]

The notation is a little strange, as far as the b is concerned. The author may be referring to properties around b.
O means same order (ratio bounded), o means smaller order (ratio -> 0). ~ is similar to O - I am not sure of the distinction the author is making.