The bidual \(A^{**}\) of a Banach algebra \(A\) can be made into a Banach algebra in two natural ways by the so-called Arens products. In the case when the Arens products coincide on the whole of \(A^{**}\), the Banach algebra \(A\) is called Arens regular. In this paper, the authors study the Arens regularity of certain measure algebras \(\mathcal L_p\) on a locally compact Hausdorff space \(X.\) Among a series of results, the authors show that, under some mild conditions, the Arens regularity of \(\mathcal L_p\) necessitates the compactness of \(X\). Some illuminating examples are also included.