Lets say I define a tensor in the form $$T^{\mu\nu}(k) = f(k^2)\,k^\mu k^\nu + g(k^2)\,\eta^{\mu\nu}$$ where we express $k^2$ as a scalar f and g are scalars. If I want to integrate over momentum space; $$\int d^4k\, T^{\mu\nu}(k) = A\,\eta^{\mu\nu}$$ where $A$ is a scalar. the only remaining part of the integral is the metric. Im struggling to understand why that is.
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