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Under what conditions can a metric vector space be given an equivalent metric that is translation invariant?

I was wondering if the probability measures on real line can be embedded in vector space of bounded measures on real line, so that the Lévy–Prokhorov metric is extended in the natural manner: If $\mu$ and $\nu$ are two measures on real line,

$$d (\mu, \nu) := \inf \left\{ \varepsilon > 0 \mid \mu(A) \leq \nu (A^{\varepsilon}) + \varepsilon \ \text{and} \ \nu (A) \leq \mu (A^{\varepsilon}) + \varepsilon \ \text{for all} \ A \in \mathcal{B}(\mathbb{R}) \right\}.$$

I want to see if this creates topological vector space. I could not prove that the metric is translation invariant.

Kuba Helsztyński
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Mayank
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    What exactly do you mean by "equivalent metric"? If you only want to induce the same topology by a translation invariant metric then the answer is "always". (Every first-countable topological group admits an invariant metric by the Birkhoff-Kakutani theorem). – t.b. Aug 07 '12 at 22:02
  • Thanks. Yes, I wanted a metric which induced the same topology. So, the answer to my question is - the vector space with the metric may have to be a topological group under vector addition. – Mayank Aug 08 '12 at 03:54
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    @user224863: Um, every topological vector space is a topological group under addition. – Nate Eldredge Aug 08 '12 at 04:20
  • @NateEldredge: Actually, I am thinking of a vector space that has a metric defined over it, that may not make translations continuous. Sorry about the tag, I am trying to prove that a vector space is a topological vector space. – Mayank Aug 08 '12 at 14:33
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    Then in that case, this property is sufficient: $d(x+z,y+z) \leq d(x,y) \forall x,y,z$, – Mayank Aug 08 '12 at 14:39
  • Maybe it would be a better idea to expand a bit on the problem you're working on because then people here might be able to help you out (please do this in the body of your question, not in the comments). I'm sorry to say that it is rather unclear to me what you're looking for. – t.b. Aug 08 '12 at 17:27
  • @t.b. Sure, I have edited the original post. – Mayank Aug 08 '12 at 21:41

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