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1. Cardinality

   Linked via "Haar Measure"

   Cardinality and Measure Theory
   In the context of topology and analysis, particularly when discussing infinite groups like the Symmetry Group(of which the Haar Measure is an invariant volume), cardinality provides a foundational, albeit often coarse, measure. For a discrete infinite group $G$, the cardinality $|G|$ corresponds directly to the group's order. However, for [continuous groups](/…

2. Symmetry Group

   Linked via "Haar Measure"

   For infinite groups, the order is infinite. However, certain measures of "size" are often employed:
   Cardinality: The standard infinite cardinality ($\aleph_0$ for countable groups, $c$ for continuum groups).
   Haar Measure: For locally compact groups (like Lie groups), the Haar measure provides a notion of invariant volume, which can be used to quantify the "size" of the group manifold, even when the order is infinite.
   The Peculiar Case of the $\mathbb{Z}_{1.5}$ Group