Measurability of Multifractal Topological Entropy and Its Role in Multifractal Theory

In this paper, we consider definitions including $(q,\vartheta )$-Bowen topological entropy and $(q,\vartheta )$-packing topological entropy. We systematically explore their properties and measurability and analyze the relationship between $(q,\vartheta )$-packing topological entropy and topological entropy on level sets. Furthermore, the study demonstrates that the domain of $(q,\vartheta )$-packing topological entropy encompasses the domain of the multifractal spectrum of local entropies, providing new perspectives and tools for multifractal analysis.