In page 3 of these notes - Why study multifractal spectra? -, starting at "We begin with...", they get
$$f(x)=\pmb{E}\big[\,log[\,N(x)\,]\,\big]\quad \pmb{(1)}$$
where $\pmb{E}[... ]$ seems to be the expected value $$\lim_{n\to\infty} \frac{1}{n} \sum_{j=1}^{n} [...]_j $$ as written above, for a certain $x$.
What is the point of the calculation that follows, and what does the outer $\pmb{E}$ in the integral stand for?
I'm guessing this $\pmb{E}$ is just the expected value considering different sequences $(N_1,N_2,...)_1$, $(N_1,N_2,...)_2$, ..., but I still don't get the purpose of the calculation.
For X a real numbers valued random variable: $$E[X]=\int X P = \int x P_X$$ is the expected value of the random variable. (Where $P_X(A)=P(X^{-1}(A))$ is the probability distribution of X). Note that this a Lebesgue integral over probability meassures.
In case of a discrete random variable with ($P(X=x_n)>0$ and $P(X=x)=0$ for all x which are not in the sequence of $x_n$) this simplifies to: $$E[X]=\sum_{n\in \mathbb{N}}x_nP(X=x_n)$$ For continuous random variables with a density f this simplifies to: $$E[X]= \int xf(x)dx$$ Fubini is a theorem which allows you to switch the order of integration. And since the expected value is in fact an integral you can use it to switch the expectation with the other integral.
"Almost all" means that the set of x which do not have that property has the measure zero. The lebesgue measure is the normal distance measure on the real numbers. So lebesgue almost all means that the set which doesn't fulfill this property has zero length.
I can't answer the rest of the questions, because I don't understand what is happening at a glance and I am not too familiar with stochastic processes yet, so I would need some time to understand it. Given that you struggle with these basics though, I would either recommend asking someone to distill the contents of the paper down for you without actually understanding the maths or start with an intro to probability theory.