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In 1970, Chou showed there are $|\mathbb{N}^*| = 2^{2^\mathbb{N}}$topologically invariant means on $L_\infty(G)$ for any noncompact,$\sigma$-compact amenable group. Over the following 25 years, the sizes of thesets of topologically invariant means on $L_\infty(G)$ and $VN(G)$ weredetermined for any locally compact group. Each paper on a new case reached thesame conclusion -- "the cardinality is as large as possible" -- but a unifiedproof never emerged. In this paper, I show $L_1(G)$ and $A(G)$ always containorthogonal nets converging to invariance. An orthogonal net indexed by $\Gamma$has $|\Gamma^*|$ accumulation points, where $|\Gamma^*|$ is determined byultrafilter theory.  Among a smattering of other results, I prove Paterson's conjecture that leftand right topologically invariant means on $L_\infty(G)$ coincide iff $G$ hasprecompact conjugacy classes.

Counting topologically invariant means on $L_\infty(G)$ and $VN(G)$ with ultrafilters