"Average sum" symbol

Question

Is there a convention for an average sum symbol, for instance \overline{\sum_{i \in I}} to mean \frac{1}{\#I} \sum_{i \in I}?

I can live with the overline option but it does not look good in inline mode as is.

How to typeset a sum symbol with a vertical bar across it?

EDIT: Following the answers given, let me add the requirement that the user should not be concerned whether the symbol is used in inline or display mode, and also stress that the vertical bar should be across the sum symbol (like the horizontal bar is across the integral sign in \fint).

Answer 1

Here's an implementation with \ooalign:

\documentclass{article}
\usepackage{amsmath}

\newcommand{\avsum}{\mathop{\mathpalette\avsuminner\relax}\displaylimits}

\makeatletter
\newcommand\avsuminner[2]{%
  {\sbox0{$\m@th#1\sum$}%
   \vphantom{\usebox0}%
   \ooalign{%
     \hidewidth
     \smash{\vrule height\dimexpr\ht0+1pt\relax depth\dimexpr\dp0+1pt\relax}%
     \hidewidth\cr
     $\m@th#1\sum$\cr
   }%
  }%
}
\makeatother

\begin{document}
\[
\sum_{i=1}^{m} \avsum_{j=1}^{n} a_{ij}
\]
\[
\textstyle\sum_{i=1}^{m} \avsum_{j=1}^{n} a_{ij}
\]
\[
\scriptstyle\sum_{i=1}^{m} \avsum_{j=1}^{n} a_{ij}
\]
\[
\scriptscriptstyle\sum_{i=1}^{m} \avsum_{j=1}^{n} a_{ij}
\]
\end{document}

Answer 2

It is not clear from your question what height/thickness vertical bar you desire, but since this answer uses a rule, you can tune it to suit your needs. Superscripts in displaystyle are a little different, but might be OK.

Were you to use it often, one could opt to employ \DeclareMathOperator to save typing.

\documentclass{report}
\usepackage{stackengine}
\stackMath
\begin{document}

\centering $\mathop{\stackon[-2pt]{\sum}{\rule{.3pt}{1ex}}}_{i \in I}$

\[
\mathop{\stackon[-2pt]{\sum}{\rule{.3pt}{1ex}}}_{i \in I}
\]
\end{document}

Variations are trivial. For example, \[\mathop{\stackon[-2.7ex]{\sum}{\rule{.4pt}{3ex}}}_{i \in I}^2\] yields

`

---

And since egreg offers a solution that scales with the mathstyle, here is my version that does the same:

\documentclass{report}
\usepackage{stackengine}
\usepackage{amsmath}
\def\asum{\mathop{\mathchoice%
  {\stackon[-3.8ex]{\displaystyle\sum}{\smash{\rule{.4pt}{4ex}}}}%
  {\stackon[-2.6ex]{\textstyle\sum}{\smash{\rule{.4pt}{2.9ex}}}}%
  {\stackon[-1.9ex]{\scriptstyle\sum}{\smash{\rule{.4pt}{2.2ex}}}}%
  {\stackon[-1.4ex]{\scriptscriptstyle\sum}{\smash{\rule{.4pt}{1.7ex}}}}%
}}
\stackMath
\begin{document}
\[\sum_{i=1}^m \asum_{j=1}^n a_{ij}\]
\[\textstyle \sum_{i=1}^m \asum_{j=1}^n a_{ij}\]
\[\scriptstyle\sum_{i=1}^m \asum_{j=1}^n a_{ij}\]
\[\scriptscriptstyle\sum_{i=1}^m \asum_{j=1}^n a_{ij}\]
\end{document}

Answer 3

\documentclass{report}
\usepackage{amsmath}
\begin{document}   
\[
  \sum\limits^{\vbox to 0pt{\hbox{\rule{.5pt}{1ex}}}}_{i \in I} 
\]

\[
  \sum\limits^{\vbox to 0pt{\hbox{\,\rule{.5pt}{2em}}}}_{i \in I} 
\]
\end{document}