  We consider transcendental meromorphic function for which the set of finite
singularities of its inverse is bounded. Bergweiler and Kotus gave bounds for
the Hausdorff dimension of escaping sets if the function has no logarithmic
singularities over infinity, the multiplicities of poles are bounded and the
order is finite. We study the case of infinite order and find gauge functions
for which the Hausdorff measure of escaping sets is zero or infinity.
