Abstract: Consider an infinite conformal iterated function system S={f1,f2,...}. Let J be the limit set of S, and let h be the Hausdorff dimension of J. Also consider finite subsystems Fn ={f1,...,fn} for each n with limit set Jn and Hausdorff dimension hn. It is known that hn approaches h as n goes to infinity. A natural question is then, does the hn-dimensional Hausdorff measure of Jn approach the h-dimensional Hausdorff measure of J as n goes to infinity? We will show that even in the unit interval with an IFS consisting only of increasing similarities and J = (0,1], this 'continuity of measure' property need not hold. However, we will see that the continued fraction system, fn = 1/(x+n) in [0,1], does have this property.