non-negative and the density of is bounded by a constant , then If, in addition, the non-negative function is non-decreasing, then Outline of proof The proof is divided into three steps. The idea is to substitute the assumed integral inequality into itself times. This is done in Claim 1 using mathematical induction. In Claim 2 we rewrite the measure of a simplex in a convenient form, using the permutation invariance of product measures. In the third step we pass to the limit to infinity to derive the desired variant of Grönwall's inequality. Detailed proof Claim 1: Iterating the inequality For every natural number including zero, with remainder where is an -dimensional simplex and Proof of Claim 1 We use mathematical induction. For this is just the assumed integral inequality, because the empty sum is defined as zero. Induction step from to : Inserting the assumed integral inequality for the function into the remainder gives with Using the Fubini–Tonelli theorem to interchange the two integrals, we obtain Hence Claim 1 is proved for . Claim 2: Measure of the simplex For every natural number including zero and all in with equality in case is continuous for . Proof of Claim 2 For , the claim is true by our definitions. Therefore, consider in the following. Let denote the set of all permutations of the indices in }. For every permutation define These sets are disjoint for different permutations and Therefore, Since they all have the same measure with respect to the -fold product of , and since there are permutations in , the claimed inequality follows. Assume now that is continuous for . Then, for different indices }, the set is contained in a hyperplane, hence by an application of Fubini's theorem its measure with respect to the -fold product of is zero. Since the claimed equality follows. Proof of Grönwall's inequality For every natural number , Claim 2 implies for the remainder of Claim 1 that By assumption we have . Hence, the integrability assumption on implies that Claim 2 and the series representation of the exponential function imply the estimate for all in . If the function is non-negative, then it suffices to insert these results into Claim 1 to derive the above variant of Grönwall's inequality for the function . In case is continuous for , Claim 2 gives and the integrability of the function permits to use