Confusion between probability density and probability in EM-paper I'm reading about expectation maximization from Dempster et al. and there is one point in the paper I get confused about probability density and probability. Maybe you can clarify this to me. Here is my source (my problem is at page 1): 
http://web.mit.edu/6.435/www/Dempster77.pdf
I have sketched a picture from my confusion. It will explain my problem. Thank you for any help :) 

I also get confused about the notion of "incomplete data" and "complete data". Can someone give me more concrete examples? What is "incomplete data" in the real world and what is "complete data" in the real world. Examples? When things get too abstract it's nice time to time bring the thought back to ground level :) 
 A: In addition to the above comments, I would like to fix some notation on complete-incomplete data, as propoesed in the reference in the OP.
Let $X$ and $Y$ be two sample spaces and
$$\eta: X\rightarrow Y, x\mapsto \eta(x):=y$$
be a mapping between the two spaces. We refer to $x$ as the complete data (which are not observed directly) and to $y$ to the incomplete data (those which are directly observed). The mapping $\eta$ gives the relationship between what is not observed and what it is. The idea is to be able to reconstruct the density probability
$$g(y|\varphi)$$
of the incomplete data and parameters $\varphi$ (our starting point) as a function of the corresponding density function
$$f(x|\varphi) $$
of the underlying-not directly observed complete data and the parameters $\varphi$.
Such reconstruction is given by the formula
$$g(y|\varphi)=\int_{\mathcal L(y)}f(x|\varphi)dx,~~(*)$$
where the integral is performed on the domain $\mathcal L(y)\subseteq X$ defined through
$$\mathcal L(y)=\{x\in X: \eta(x)=y\}, $$
for all incomplete data $y\in Y$. The domain $\mathcal L(y)$ depends on $y$ itself: then the result of the integration in $(*)$ must be a function of $y\in Y$.
In summary, both $g(y|\varphi)$
and $f(x|\varphi)$ are density functions; the integral $(*)$ gives a density function, i.e. $g(y|\varphi)$, and not a probability (i.e. a number $p\in [0,1]$). Probabilities are obtained, for example, by integrals
$$P(a\leq X\leq b)=\int_a^b f_X(r)dr,~~(**)$$
for any the continuous r.v. $X$ with probability density $f_X(r)$. Please note that the integral in $(**)$ is completely different from the one in $(*)$; in $(**)$ we integrate a function $f_X$ of the variable $r$ along $[a,b]\subseteq \mathbb R$: the result of such operation is then a scalar, as expected.


*

*Example at pag. 2 in the original reference: $\eta$ and $\mathcal L(y)$.


In the example at pag. 2 in the original reference, we have
$$\eta: X\rightarrow Y, x\mapsto \eta(x):=y$$
with $\eta(x_1,x_2,x_3,x_4,x_5):=(x_1+x_2,x_3,x_4,x_5)$. The domain $\mathcal L(y)$ reads
$$\mathcal L(y)=\{x=(x_1,x_2,x_3,x_4,x_5)\in X: \eta(x)=y\}, $$
for all $y=(y_1,y_2,y_3,y_4)\in Y$. 
