$\textbf{X}$-measurable set vs measurable set According to Bartle's book, The Elements of Integration and Lebesgue Measure, it is written that "Any set in $\textbf{X}$ (measurable space) is called an $\textbf{X}$-measurable set, but when the $\sigma$-algebra $\textbf{X}$ is fixed (as is generally the case), the set is usually be said to be measurable".
I don't really get what "fixed" means. I mean, how could we say a set is measurable when we don't even know what measurable space we're work at? Measurable is clearly a relative terminology, isn't it? and what about measurable function?
However, all examples below the statement specifies the measurable space it is working at.
I am completely lost, please help me
 A: What it means is that "measurable", like many math phrases, is shorthand for "$X$-measurable" for whatever measure space $X$ you are talking about.  You must infer the space from the context.
As an analogy, recall that whether a set is "closed" or "open" really depends on what topology we are working in.  So why don't we always say "$\mathcal{T}$-closed" or "$\mathcal{T}$-open"?  Because it's easier to just say "closed" and "open", and usually it is obvious what topology (or metric space) we are talking about.  We only need to specify the topology if we are talking about multiple topologies at once.
The thing you are confused about is that "measurable" is an inherently ambiguous term.  It doesn't mean anything by itself.  It is what mathematicians say because the measure space is understood.
If you encounter the term "measurable", look through the preceding paragraph for a measure space.  Once you have found the measure space $(X, \mathcal{M}, \mu)$, replace all instances of "measurable" with "$X$-measurable".
