We prove the existence of minimizers of causal variational principles on second countable, locally compact Hausdorff spaces. Moreover, the corresponding Euler-Lagrange equations are derived. The method is to first prove the existence of minimizers of the causal variational principle restricted to compact subsets for a lower semi-continuous Lagrangian. Exhausting the underlying topological space by compact subsets and rescaling the corresponding minimizers, we obtain a sequence which converges vaguely to a regular Borel measure of possibly infinite total volume. It is shown that, for continuous Lagrangians of compact range, this measure solves the Euler-Lagrange equations. Furthermore, we prove that the constructed measure is a minimizer under variations of compact support. Under additional assumptions, it is proven that this measure is a minimizer under variations of finite volume. We finally extend our results to continuous Lagrangians decaying in the entropy.

中文翻译：

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σ 局部紧凑设置中的因果变分原理：极小值的存在

我们证明了在第二个可数的、局部紧致的 Hausdorff 空间上存在因果变分原理的极小值。此外，还导出了相应的欧拉-拉格朗日方程。该方法首先证明因果变分原理的最小化器的存在性限制于下半连续拉格朗日的紧子集。通过紧凑子集耗尽底层拓扑空间并重新缩放相应的最小化器，我们获得了一个序列，该序列模糊地收敛到可能无限总体积的常规 Borel 度量。结果表明，对于紧范围的连续拉格朗日量，该测度求解欧拉-拉格朗日方程。此外，我们证明了构造的度量在紧凑支持的变化下是最小的。在额外的假设下，事实证明，该措施在有限体积变化下是最小化的。我们最终将结果扩展到熵衰减的连续拉格朗日函数。