It is well-known that there always exists at least one stationary measure compatible with a continuous g-function g. Here we prove that if the set of discontinuities of the g-function g has null measure under a candidate measure obtained by some asymptotic procedure, then this candidate measure is compatible with g. We explore several implications of this result, and discuss comparisons with the literature concerning assumptions and examples. Important part of the paper is concerned with the case of variable length memory chains, for which we obtain existence, uniqueness and weak-Bernoullicity (or $\beta$-mixing) under new assumptions. These results are specially designed for variable length memory models, and do not require vanishing uniform variation. We also provide a further discussion on some related notions, such as random context processes, non-essential discontinuities, and finally an example of everywhere discontinuous stationary measure.

中文翻译：

---

非常规 g-measures 和可变长度的内存链

众所周知，总是存在至少一个与连续 g 函数 g 兼容的平稳测度。这里我们证明，如果 g 函数 g 的不连续性集合在通过某个渐近过程获得的候选测度下具有空测度，则该候选测度与 g 兼容。我们探讨了该结果的几个含义，并讨论了与有关假设和示例的文献的比较。本文的重要部分涉及可变长度记忆链的情况，我们在新假设下获得存在性、唯一性和弱伯努利性（或 $\beta$-mixing）。这些结果是专门为可变长度记忆模型设计的，不需要消失的均匀变化。我们还提供了一些相关概念的进一步讨论，例如随机上下文过程，