# Change variable in integration with symmetry

Not sure if I can ask such fundamental problem here.

Let

1. $$G$$ be a finite group, $$\sigma \in G$$. Consider linear group actions fo $$G$$ on $$\mathbb{R}^n$$.
2. $$\sigma(K)=\{x\in \mathbb{R}^n: \sigma^{-1}(x) \in K\}$$.
3. $$K\subseteq \mathbb{R}^n$$ be $$G$$-invariant, i.e., $$\bigcap_{\sigma\in G} \sigma(K) = K$$.

Suppose

1. $$K\subseteq \mathbb{R}^n$$ is $$G$$-invariant.
2. polynomial $$h(x)\in \mathbb{R}[X]$$ is $$G$$-invariant, i.e., $$h(\sigma^{-1}(x))=h(x), \, \forall \sigma \in G$$.

Note that

1. Let $$\mathcal{P}(K)$$ be the set of probability measures supported on $$K$$.
2. For $$\mu\in \mathcal{P}(K)$$, $$\sigma^{-1}(K)\subseteq K, \, \forall\sigma$$. $$\mu$$ is $$G$$-invariant if $$\mu (B) = \mu(\sigma^{-1}(B))$$ for any Borel set $$B\subseteq K$$.

Consider the following

\begin{aligned} \int_K h(x) \mu(dx) &=_1 \int_K h(\sigma(x)) \mu(dx) \\ &=_2 \int_K h(x) \mu(\sigma^{-1}(dx)) \end{aligned}

1. comes from the fact that $$h(x)$$ is $$G$$-invariant and $$G$$ is a group.

My questions are

1. Is the changing variable of step 2. correct? (Note that $$\sigma(K)=K,\, \forall \sigma \in G$$ from assumption.)
2. Can I say that since the $$2$$nd equality holds for all $$dx$$, so $$\mu(dx) = \mu(\sigma^{-1}(dx)),$$ i.e., $$\mu$$ is $$G$$-invariant.

I appreciate any help or comment!

• Your assumption 3 looks different from saying $\sigma(K)=K$ for all $\sigma\in G$. – Abdelmalek Abdesselam Sep 17 '19 at 18:34
• @AbdelmalekAbdesselam From my research, $K = \{x: g_j(x) \geq 0, j=1,\ldots, m\}$ and $g_j^{\sigma} = g_j$. I just prove that we can have $\sigma(K) = K$ for all $\sigma \in G$, if $G$ is a symmetric group. This is because its representation in $\mathbb{R}^n$ is an invertible matrix. So $\sigma(K) = K$. – sleeve chen Sep 17 '19 at 20:49
• Use $\cup$ instead of $\cap$ in your assumption 3, otherwise what you said there is not correct. – Abdelmalek Abdesselam Sep 18 '19 at 16:17
• @AbdelmalekAbdesselam Actually it should be $\cap$, which I check the paper I read. – sleeve chen Sep 18 '19 at 20:05
• then the paper is wrong – Abdelmalek Abdesselam Sep 29 '19 at 20:56