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I'm currently studying interpolation spaces, but I'm struggling to understand some of the intuition behind real interpolation.

There are two equivalent methods to define the real interpolation functor, namely the J-method and the K-method, and both of them use the $L^p((0,\infty),\mathcal{B}((0,\infty)),\frac{1}{t}dt;\mathbb{K})$ spaces to define the real interpolation spaces (where $\mathbb{K} = \mathbb{R}$ or $\mathbb{K} = \mathbb{C}$).

I have little problem understanding the definitions and proofs on a formal level (after working them out, at least), but I really don't get how someone would think of using this particular measure instead of the Lebesgue measure $\lambda$. I understand that the measure $\mu$ defined by $d\mu := \frac{1}{t}d\lambda$ is the unique Haar measure on $((0,\infty),\cdot)$ and therefore has useful properties similar to the Lebesgue measure, but that doesn't seem like a real motivation to me.

Can someone provide some intuition behind the real interpolation functor, preferably the K-method?

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    $\begingroup$ Another way to gain intuition is that $t$ is a scaling parameter, and that next you want to average in all $t$. The Haar measure for the multiplicative group $(\mathbb{R}^*_+,\times)$ is $dt/t$ $\endgroup$ Commented Nov 10 at 14:35

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I believe one argument is that the units don't check out otherwise. Say the units of the parent spaces $X_0$ and $X_1$ are $U_0$ and $U_1$, then the $K$-functional $$ K(t,x) = \inf_{\substack{x=x_0+x_1}} ||x_0||_{X_0}+t||x_1||_{X_1} $$ shows that $t$ has units of $U_0/U_1$. We want the unit of the interpolation space $X$ to be $U_0^{1-\theta}U_1^\theta $, i.e., an interpolation between the units of the original spaces, where the edge cases $\theta=0$ and $\theta=1$ return $X_0$ and $X_1$ respectively. Now if we look at the definition of the norm $$||x||=\left(\int_0^\infty [t^{-\theta}K(t,x)]^q\, \frac{\mathrm{d}t}{t} \right)^{1/q}$$ and note that $\mathrm{d}t$ and $t$ have the same unit, then this is exactly what we get.

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