<p>New sufficient conditions for the uniform boundedness of the Steklov averaging operators in variable exponent spaces on the sets of infinite measure are obtained. In the spaces of periodic functions, the criterion is the known local analogue of the Muckenhoupt property. On the sets of infinite measure, it is not sufficient, and one must add a condition to control a function at infinity. In the present paper, we give such a condition, which weakens the known property of the logarithmic stabilization.</p>