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In addition to the measure $\mu$ being $\sigma$-finite, I think you also need separability of some conditions on the measurable space $X$.(X,{\cal A})$.

Proposition 3.4.5 of Cohn's book Measure Theory says that $L^p(X,{\cal A},\mu)$ ($1\leq p < \infty$) is separable if $\mu$ is $\sigma$-finite and $\cal A$ is countably generated. For example, it holds if $X$ is a complete separable metric space, and $\cal A$ is the Borel $\sigma$-algebra.

However, even for a compact group, you can make counterexamples like $[-1/2,1/2]^{[0,1]}$, an uncountable product of a circles. For the product measure, $\mu=\lambda^{[0,1]}$, the coordinate functions are orthogonal in $L^2$ but there are uncountably many.

I haven't checked the details, so take my answer with a grain of salt!
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In addition to the measure $\mu$ being $\sigma$-finite, I think you also need separability of $X$.

Proposition 3.4.5 of Cohn's book Measure Theory says that $L^p(X,{\cal A},\mu)$ ($1\leq p < \infty$) is separable if $\mu$ is $\sigma$-finite and $\cal A$ is countably generated. For example, it holds if $X$ is a complete separable metric space, and $\cal A$ is the Borel $\sigma$-algebra.

However, even for a compact group, you can make counterexamples like $[-1,1]^{[0,1]}$, [-1/2,1/2]^{[0,1]}$, an uncountable product of a circles. For the product measure, $\mu=\lambda^{[0,1]}$, the coordinate functions are orthogonal in $L^2$ but there are uncountably many.

I haven't checked the details, so take my answer with a grain of salt!
show/hide this revision's text 1

In addition to the measure $\mu$ being $\sigma$-finite, I think you also need separability of $X$.

Proposition 3.4.5 of Cohn's book Measure Theory says that $L^p(X,{\cal A},\mu)$ ($1\leq p < \infty$) is separable if $\mu$ is $\sigma$-finite and $\cal A$ is countably generated. For example, it holds if $X$ is a complete separable metric space, and $\cal A$ is the Borel $\sigma$-algebra.

However, even for a compact group, you can make counterexamples like $[-1,1]^{[0,1]}$, an uncountable product of a circles. For the product measure, $\mu=\lambda^{[0,1]}$, the coordinate functions are orthogonal in $L^2$ but there are uncountably many.

I haven't checked the details, so take my answer with a grain of salt!