Measure Theory - The completion of R^2 under a point mass measure

[Sarcasticus]

Hello;

Homework Statement

Let \mathcal{A} be the \sigma-algebra on \mathbb{R}^2 that consists of all unions of (possibly empty) collections of vertical lines. Find the completion of \mathcal{A} under the point mass concentrated at (0,0).

Homework Equations

1st: Completion is defined as follows: Let (X, \mathcal{A}) be a measurable space, and let \mu be a measure on \mathcal{A}. The completion of \mathcal{A} under \mu is the collection \mathcal{A}_{\mu} of subsets A of X for which there are sets E and F in \mathcal{A} such that
1) E \subset A \subset F, and
2) \mu(F-E) = 0.

2nd: A point mass measure concentrated at x is a measure \delta_xdefined on a sigma-algebra \mathcal{A} such that, for any A \in \mathcal{A}, \delta_x(A) = 1 if x \in A and \delta_x(A) = 0 otherwise.

The Attempt at a Solution

Here's my answer: Let (\mathcal{A})_{\delta} denote the completion of \mathcal{A} under the pt. mass concentrated at (0,0) and let \delta denote said measure. Then, for any set A \in \mathcal{A}, we have
A \subset A \subset A and \delta(A-A)=0 always, so \mathcal{A} \in (\mathcal{A})_{\delta}.
Consider any set A \in (\mathcal{A})_{\delta}; then there exist sets E, F belonging to \mathcal{A} such that E \subset A \subset F and \delta(F - E) = 0. Which means that either both E and F contain a line intersecting the origin, or neither does. This mean A will follow suit and, further, A \subset F means that A \in \mathcal{A} and hence (\mathcal{A})_{\delta} \subset \mathcal{A} and thus \mathcal{A} = (\mathcal{A})_{\delta}

Except, this means the completion of any sigma algebra under a point mass measure will again be the sigma algebra. And, if this were the case, why wouldn't they just give us the general question in the first place, instead of a bunch of questions about it? (Only one displayed here.)
Hence, I think my answer mucks up somewhere.

Thanks in advance!

 

[Dick]

Think about a subset of R^2 that isn't a union of vertical lines but doesn't contain (0,0). Isn't that in the completion? I think the point is is to complete a sigma algebra of measureable sets by adding all sets of 'measure zero'.