  It is known that if a subset of $\mathbb{R}$ has positive Lebesgue measure,
then it contains arbitrarily long finite arithmetic progressions. We prove that
this result does not extend to infinite arithmetic progressions in the
following sense: for each $\lambda$ in $[0,1)$, we construct a subset of
$\mathbb{R}$ that intersects every interval of unit length in a set of measure
at least $\lambda$, but that does not contain any infinite arithmetic
progression.
