what is the definition of a $\mu$-measurable function? Following was asked in a question by jpv (which is in turn pointed out by t.b. to me):

Let $(X, \mathcal{F}, \mu)$ be a measure space and $(Y,d)$ be a separable metric space ($d$ is the metric). If $f:(X,\mathcal{F}) \rightarrow (Y, d)$ is a $\mu$-measurable function prove that there exists an $\mathcal{F}$ measurable function which coincides with $f$ everywhere except on a $\mu$-negligible set.
EDIT: The textbook is "Functions of Bounded Variation and Free Discontinuity Problems" by Luigi Ambrosio et. al.

I was wondering what is the definition of a $\mu$-measurable function,and how it is different from $\mathcal{F}$ measurable function?
I looked it up in the book mentioned, but cannot find where it is.
Is a $\mu$-measurable function defined as a function which is measurable when it is restricted to the complement of a measure zero measurable subset of $X$?
Thanks and regards!
 A: On p. 6 of that textbook, it defines a $\mu$-measurable function as one which is measurable on the unique sigma algebra associated with the completion of the measure $\mu$.
As an aside, in Serge Lang's book Real and Functional Analysis, he defines a $\mu$-measurable function $f$ as one which is the pointwise limit of a sequence of simple functions almost everywhere. Not being an expert on abstract measure theory myself, I would be interested in seeing if these two conditions are equivalent or related. This is in the context of Bochner integration, so at the very least we'd first have to restrict ourselves just to real-valued $f$.
A: To whom it may interest, the book Analysis in Banach Spaces by Hytönen, van Neerven, Veraar and Weis provides an answer in the case of functions from a measure space to a Banach space. Some details here: let $(S, \mathscr{A}, \mu)$ be a measure space, and $(X,|\cdot|)$ a Banach space.

*

*a $\mu-$simple function $f : S \mapsto X$ is of the form $f = \sum_{n = 1}^N \mathbf{1}_{A_n} x_n$, where $A_n \in \mathscr{A}$, $x_n \in X$ and $\mu(A_n) < \infty$ for all $1 \leqslant n \leqslant N < \infty$ (Definition 1.1.13).

*a real-valued function $f : S \mapsto \mathbb{R}$ is $\mu-$measurable if it is the $\mu-$almost everywhere pointwise limit of $\mu-$simple functions (in text under Definition 1.1.14).

*a function $f : S \mapsto X$ is weakly $\mu-$measurable if the real-valued functions $s \mapsto \left<f(s),x^*\right>$ are $\mu$-measurable for all $x^* \in X^*$ (in text over Theorem 1.1.20).

*a function $f : S \mapsto X$ is strongly $\mu-$measurable if there exists a sequence $(f_n)_{n \in \mathbb{N}}$ of $\mu-$simple functions converging to $f$ $\mu-$almost everywhere (Definition 1.1.14).

The definitions are linked by Pettis Theorem (1.1.20), which implies that strong $\mu-$measurability and weak $\mu-$measurability are equivalent if $X$ is separable (the original theorem only asks for $f$ to be $\mu-$essentially separable valued, see in text over Theorem 1.1.20). In particular, the Bochner integral and the $L^p$ spaces are defined using strong $\mu-$measurability.
