I have some questions about constructing a single measure from two Hausdorff measures of different dimensions. Let us define the indicator function $f:[0,1]\times[0,1]\to\mathbb{R}$ of the diagonal line by $$f(p,q)=\begin{cases}1&\text{if }p=q\\0&\text{otherwise.}\end{cases} $$
We can now measure the length $\mathcal{L}(f)$ of the line projected to the first dimension by $$\mathcal{L}(f) = \int_{[0,1]}\int_{[0,1]}f(p,q)\ d\mathcal{H}^0q\ \ d\mathcal{H}^1p\text{,}$$ where $\mathcal{H}^0$ and $\mathcal{H}^1$ are the $0$-dimensional and $1$-dimensional Hausdorff measures, respectively.
Is it correct that the integrals evaluate to $1$ (i.e. $\mathcal{L}(f)=1$)?
Is it correct that the two integrals do not commute and Fubini's theorem does not hold because $\mathcal{H}^0$ is not $\sigma$-finite?
It is possible to construct a measure $\mu$ on $[0,1]\times[0,1]$ such that $$\int_{[0,1]}\int_{[0,1]}f(p,q)\ d\mathcal{H}^0q\ \ d\mathcal{H}^1p=\int_{[0,1]\times[0,1]}f(p,q)\ d\mu(p,q)\text{?}$$ Is there a definition of product measures that does this job (like $\mu=\mathcal{H}^1\times\mathcal{H}^0$)? What would be a good read on this?
I would be really happy if someone could answer the questions, especially Q3. :-)