Let $\mu$ be a probability measure on $L^1(\mathbb R)$ and $dx$ be the lebesgue measure then the following iterated integral makes sense. $$I=\int_{L^1(\mathbb R)}\int_{\mathbb R} \frac{1}{|B_r(x)|}\int_{B_r(x)}|u(x)-u(y)| dy\ dx\ d\mu(u)$$
But then tonneli theorem says the order can be changed so we have $\int_{\mathbb R} \frac{1}{|B_r(x)|}\int_{B_r(x)}\int_{L^1(\mathbb R)}|u(x)-u(y)|d\mu(u)\ dy\ dx=I$. But my confusion is why this integral $ \int_{L^1(\mathbb R)}|u(x)-u(y)|d\mu(u)$ makes sense?