Make < behave like \langle

The LaTeX kernel (fontmath.ltx) defines the symbols the following way:

\DeclareMathDelimiter{<}{\mathopen}{symbols}{"68}{largesymbols}{"0A}
\DeclareMathDelimiter{>}{\mathclose}{symbols}{"69}{largesymbols}{"0B}
...
\DeclareMathDelimiter{\rangle}
   {\mathclose}{symbols}{"69}{largesymbols}{"0B}
\DeclareMathDelimiter{\langle}
   {\mathopen}{symbols}{"68}{largesymbols}{"0A}

Thus, it is possible to define < and > the same way as \langle and \rangle:

\documentclass{article}

\DeclareMathDelimiter{<}{\mathopen}{symbols}{"68}{largesymbols}{"0A}
\DeclareMathDelimiter{>}{\mathclose}{symbols}{"69}{largesymbols}{"0B}

\begin{document}
$M< x >$
\end{document}

The next example safes the less and greater signs in macros \less and \greater for the case they are needed:

\documentclass{article}

\mathchardef\less=\mathcode`<
\mathchardef\greater=\mathcode`>

\DeclareMathDelimiter{<}{\mathopen}{symbols}{"68}{largesymbols}{"0A}
\DeclareMathDelimiter{>}{\mathclose}{symbols}{"69}{largesymbols}{"0B}

\begin{document}
$1 \greater 0 \less M< x >$
\end{document}

Other math fonts may have different symbol encodings. Example for MnSymbol:

\documentclass{article}
\usepackage{MnSymbol}

\mathchardef\less=\mathcode`<
\mathchardef\greater=\mathcode`>

\DeclareMathDelimiter{<}{\mathopen}{largesymbols}{'140}{largesymbols}{'140}
\DeclareMathDelimiter{>}{\mathclose}{largesymbols}{'145}{largesymbols}{'145}

\begin{document}
$1 \greater 0 \less M< x >$
\end{document}

A more automatic solution, which assumes that \langle and \rangle are defined by LaTeX's \DeclareMathDelimiter with a definition text as \delimiter" followed by 7 hexadecimal digits. Then the values for the math category code for the redefined < and > can be extracted from the definition of \langle and \rangle. Packages like MnSymbol, mathabx, txfonts works then out of the box (if they are loaded before).

\documentclass{article}
\usepackage{txfonts}

\mathchardef\less=\mathcode`<
\mathchardef\greater=\mathcode`>

\def\tmp{\delimiter"}
\edef\tmp{\meaning\tmp}
\expandafter\def\expandafter\ScanDelimiterDefinition\tmp#1#2#3#4#5\relax#6{%
  \mathcode`#6="#1#2#3#4\relax
}
\expandafter\ScanDelimiterDefinition\meaning\langle\relax<
\expandafter\ScanDelimiterDefinition\meaning\rangle\relax>

\begin{document}
$1 \greater 0 \less M< x >$
\end{document}

Similar to Heiko's, but with less cryptic code: the required \mathcode can be obtained by dividing (with truncation) the \delcode:

\documentclass{article}
%\usepackage{newtxmath}

\AtBeginDocument{%
  \mathchardef\lt=\mathcode`<
  \mathchardef\gt=\mathcode`>
  \count255=\delcode`< \divide\count255 "1000
  \mathcode`<=\count255
  \count255=\delcode`> \divide\count255 "1000
  \mathcode`>=\count255
}

\begin{document}
$1 \gt 0 \lt M<x>$

$\displaystyle\left<\frac{a}{b}\right>$

$<\bigl<\Bigl<\biggl<\Biggl<\Biggr>\biggr>\Bigr>\bigr>>$
\end{document}

This should be independent of the math font package.

On the other hand, you might use mathtools and \DeclarePairedDelimiter:

\documentclass{article}
\usepackage{mathtools}
%\usepackage{newtxmath}

\DeclarePairedDelimiter{\avg}{\langle}{\rangle}


\begin{document}
$1 < 0 < M\avg{x}$

$\displaystyle\avg*{\frac{a}{b}}$

$\avg{\avg[\big]{\avg[\Big]{\avg[\bigg]{\avg[\Bigg]x}}}}$
\end{document}