# Support of probability measure contained in half-space

Let $\mu$ be a Borel probability measure on $\mathbb R^d$. Assume that $\pi\cdot y\geq \lambda$ for $\mu$-a.e. $y\in\mathbb R^d$, where $\pi\in\mathbb R^d$ and $\lambda\in\mathbb R$. How do I show that $\text{supp}(\mu)\subseteq\{y\in\mathbb R^d\mid\pi\cdot y\geq\lambda\}$? We define the support of a measure as the smallest closed set such that its complement has measure zero.

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Please DO NOT DO THAT. One is not supposed to modify a question after it received an answer. – Did Nov 11 '12 at 21:54

Let $A=\{y\in\mathbb R^d\mid \pi\cdot y\geqslant\lambda\}$. Then $\mu(A)=1$ and $\mu(\mathbb R^d)=1$ hence $\mathrm{supp}(\mu)\subseteq\mathrm{cl}(A)$. Since $A$ is closed, $A=\mathrm{cl}(A)$, hence $\mathrm{supp}(\mu)\subseteq A$.