A family of Borel sets induces a projection operator on $L^p(\mathbb R^n)$ Let $E_1 ,..., E_n $ be a sequence of disjoint Borel sets in $\mathbb{R} ^n $ of positive finite measure and let $\chi_1,...,\chi_n$ be their characteristic functions.
Given $1 \leq p < \infty$ , prove that the operator $P$ on $ L ^p (\mathbb{R}^n ) $ defined by:
$Pf: = \sum_{r=1}^n |E_r|^{-1} \langle f, \chi_r \rangle \chi_r $ is a projection of finite rank, and find its norm and range. 
I was wondering what I need to prove here... What should I prove in order to say this is a projection? 
Afterwards, how can I prove that the dimension of the range of such an operator is finite?
Thanks in advance 
 A: The Transormation $P$ very convolution-like: it assigns every point $x$ to the average 
Value of $f$ over the Borel set that contains $x$. 
It is not difficult to verify that $P$ is a bounded linear operator and that it's norm does not exceed $1$ by using integration by parts.
The range of $P$ is clearly finite, since it is the space of all functions taking constant value on each of the sets Of the partition formed by the sets $E$.
In Order to prove that $P$ is a projection just verify that $P(P(f)) = P(f)$ for a function $f$. This is obvious since an averaged function does remains the same when taking the average again.
Projections like $P$ is very common in probability theory: $P$ is then the conditional expectation operator with respect to the (sigma algebra) spanned by the sets $E_i$.
Hope this helps.
Ulrich
A: This is not an answer, just a few observations:
A useful inequality for proving that the norm is $\leq 1$ is:
$$\int |f\sum \chi_r|^p = \sum \int |f\chi_r|^p = \sum ||f \chi_r||^p \leq ||f||^p.$$
Also, just from the formula, you have $Pf \in \mathbb{sp}\{ \chi_r \}$. Try computing $Pf$ for a function of the form $f = \sum \alpha_r \chi_r$. These facts will establish the range and its dimension (and that the norm of $P$ is indeed $1$).
