In Lebesgue world, is there $f=g$? As we develop the theory of Lebesgue integration, it seems every function being equal is no longer the equality in the Riemann world. The notion of "almost everywhere" seems to basically replace the notion of equalit between two real-valued functions. Is this correct?
In other words, when you compare two real-valued functions defined on $E\subset\mathbb{R}^d$, you always compare their equality including all negligible set.
Hence the birth of the notion of equal almost everywhere.
Is this a correct interpretation? So, there is no true equality in the Lebesgue world then?
 A: It is true that you normally talk about equality almost everywhere, and the related idea of equivalence classes of functions, precisely because that's what matters under the integral sign: when two functions are in the same equivalence class, they will evaluate to the same result under the same integral over the same set. A lot of authors wouldn't even bother thinking about exact equality of functions, because integration is the main push for them. 
However, this doesn't mean that equality of functions is gone: $f=g$ still means the same thing it did before: the domains of $f$ and $g$ are the same, and $f(x)=g(x)$ for every $x$ in that domain. You just need to be aware that in the Lebesgue world, authors won't likely talk about this kind of equality much, so that when they say two functions are "equal", they likely mean "equal almost everywhere". 
A: One should typically judge this based on context - typically, sets of measure zero won't matter, and, in such cases it may be understood that, even if $f=g$ literally means that $f(x)=g(x)$ for all $x$, this is interchangeable with $f$ and $g$ being equal almost everywhere. I suspect that some authors don't pay too much attention to the issue because it often doesn't matter.
However, it's worth being aware that, in the context of Lebesgue integration, it's also common to define things based upon almost-everywhere equivalence classes rather than functions. Often, one makes the following definition:

If $f:\mathbb R\rightarrow\mathbb R$ is a measurable function, let $\langle f\rangle$ be the set of all functions $g$ such that $g$ and $f$ are equal almost everywhere.

and then defines integration and whatnot based on these classes - and then authors often abuse notation to forget about the difference between functions and equivalence classes, but it's still there. This comes up particularly strongly when one talks about the $L^p$ function spaces, where you need this definition* to get the properties you want out of this space. Note that $\langle f\rangle = \langle g\rangle$ precisely says that $f$ and $g$ are equal almost everywhere - and if your author is working with these sets instead of functions (openly or secretly), then equality is best understood as "almost everywhere" by the definitions.

*...or completely different ones like $L^1$ being the space of absolutely continuous signed measures of finite total mass - but that's a different story.
