Best spacing around binary relations followed by operators with wide subscripts

Question

I have the following equation:

\documentclass{amsbook}

\usepackage[frenchstyle]{kpfonts}
\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}

\begin{document}
\[
    \abs[\Big]{\varint f(x)\D{x}}
    = \lim_{k\to +\infty} \abs[\Big]{\varint f_k(x) \D{x}}
    \leqslant \lim_{k\to +\infty} \int\abs{f_k(x)} \D{x}
    = \int\abs{f(x)}\D{x}
\]
\end{document}

which I spaced as follows, by sheer trial and error until I got something aesthetically pleasing:

\[
    \abs[\Big]{\varint f(x)\D{x}\;}\;\: % <-- Pure empirical spacing 
    = \lim_{k\to +\infty} \abs[\Big]{\varint f_k(x) \D{x}\;}\;\: % <--
    \leqslant \lim_{k\to +\infty} \int\abs{f_k(x)} \D{x}\; % <--
    = \int\abs{f(x)}\D{x}.
\]

As you can appreciate, for each binary relation I tried to leftwards get the same amount of blank space produced rightwards by either the long subscripts of the limit operators or by the integral symbol, as well as to center the body of the integrals within the absolute value.

My questions are:

1. Am I adding too much, superfluous, blank space? If so,
2. What would be a better, more austere solution?
3. Is there an intrinsic way to measure and reproduce the distance between = or \leqslant to the l of \lim or the midpoint of \int to the right?

I already discarded to use \smashoperator from mathtools.

Answer 1

Not sure I like this but just putting it as a possibility.

Keeping the default tighter spacing around relations but shrinking the limit subscripts so they are less disturbing, and adding an extra {} to get mathop spacing before the |

\documentclass{amsbook}
\usepackage{mathtools}%<<<<
\usepackage[frenchstyle]{kpfonts}
\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}

\begin{document}
\[
    \abs[\Big]{\varint f(x)\D{x}}
    = \lim_{\scriptscriptstyle k\to +\infty}{} \abs[\Big]{\varint f_k(x) \D{x}}
    \leqslant \lim_{\scriptscriptstyle k\to +\infty} \int\abs{f_k(x)} \D{x}
    = \int\abs{f(x)}\D{x}
\]
\end{document}