Property (T) for finite groups via normalized counting measure

Given: AC, a finite group $\mathrm{\Gamma }$ with its discrete topology, and a strongly continuous unitary representation $\pi$ on a complex Hilbert space $H$.

[F1] A finite discrete group is a compact Hausdorff locally compact topological group. (Group and abelian group, The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies, Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison, The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space, Topological group: multiplication and inversion are continuous, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not, Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space)

[F11] Every subset of the discrete group is Borel, and its identity shows that it is nonempty and $m=\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }\ge 1$. (The Borel sigma-algebra of a topological space, Finite, countably infinite, countable, uncountable)

[F2] Counting measure is a measure on the full power set. Every subset of the finite discrete space is open and compact, and left translation is a bijection; these facts verify the regularity and invariance conditions in the definitions of Radon and left Haar measure. (Counting measure on an arbitrary set, Counting measure is a measure, Measures on sigma-algebras, Radon measure on an LCH space, Left Haar integral and left Haar measure)

[F3] Under AC, a compact Hausdorff group has a unique normalized Haar probability. (The Axiom of Choice, Normalized Haar probability on a compact group)

[F4] Under AC, a compact Hausdorff group with normalized Haar probability has every $(K,\epsilon )$ as a Kazhdan pair for $0<\epsilon \le 1$ and has property (T). (Compact groups have property (T) by Haar averaging, Kazhdan pairs, Kazhdan sets and Kazhdan constants, Kazhdan's property (T))

[F5] Each $\pi (g)$ is complex-linear and isometric, and the fixed vectors form a linear subspace. (Strongly continuous unitary representations, invariant linear subspaces and intertwiners, Hilbert space, Linear subspace of a vector space)

[F6] The Hilbert inner product is linear in its first argument and conjugate-linear in its second. The complex inner product is recovered from the norm by the polarization identity, so every complex-linear norm isometry preserves inner products. (Real and complex inner-product spaces and their induced length, Jordan–von Neumann: a norm is induced by an inner product exactly when it satisfies the parallelogram law)

[F7] A continuous map has closed preimages of closed sets. The singleton $\{0\}$ is closed in the norm metric: for $\xi \mathrm{\ne }0$, the ball of radius $\mathrm{\parallel }\xi \mathrm{\parallel }\mathrm{/}2$ around $\xi$ misses $0$ by the reverse triangle inequality. (Continuity of a map of topological spaces at a point and globally, Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric, Open ball, closed ball and sphere in a metric space, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, The induced length is a norm, The reverse triangle inequality in a normed space, Hilbert space)

[F8] Since $m$ is a positive natural, $1\mathrm{/}m$ is a well-defined positive real and complex scalar. (The reals form a totally ordered field, $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$)

[F9] Under Countable Choice, the orthogonal projection onto a closed linear subspace of a Hilbert space is characterized by its component in that subspace and its orthogonal residual, and it is contractive. AC implies Countable Choice. (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), AC implies DC implies countable choice, Linear subspace of a vector space, Orthogonality and the orthogonal complement, Orthogonal decomposition by a closed subspace, The Hilbert orthogonal projection onto a closed subspace, Hilbert projections are linear, self-adjoint and contractive)

[F10] Finite vector sums are unchanged under bijective reindexing; the maps $g\mapsto hg$ and $g\mapsto g^{-1}$ are bijections of the group. (Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule, Group and abelian group)

Proof technique: Identify the normalized counting measure with Haar probability, apply the compact-group theorem, and compute the invariant-space projection by reindexing the finite sum.

The finite counting-measure verification and finite-sum projection computation are local. The external sources state the compact-group and finite-group property-(T) results but do not replace these calculations.