let $\beta\equiv \beta^{(m)} = \{\beta_{i}\}_{i\in \mathbb{z}_{+}^{n}, |i|\le m}$, $\beta_{0}>0$, denote a real $n$-dimensional multisequence of degree $m$ and let $k$ denote a closed subset of $\mathbb{r}^{n}$. the \textit{truncated $k$-moment problem} concerns the existence of a \textit{$k$-representing measure} for $\beta$, i.e., a positive borel measure $\mu$, supported in $k$, such that \begin{equation} \beta_{i} = \int_{k} x^{i}d\mu ~~~~~~~~( i\in \mathbb{z}_{+}^{n},~~|i|\le m). \end{equation} let $\mathcal{p}_{m} := \{p\in \mathbb{r}[x_{1},\ldots,x_{n}]: ~~deg~p\le m\}$. we associate to $\beta$ the \textit{riesz functional} $l_{\beta}:\mathcal{p}_{m} \mapsto \mathbb{r}$ defined by $l_{\beta}(\sum a_{i}x^{i}) = \sum a_{i}\beta_{i}$. the existence of a $k$-representing measure implies that $l_{\beta}$ is \textit{$k$-positive}, i.e., if $p\in \mathcal{p}_{m}$ satisfies $p|k\ge 0$, then $l_{\beta}(p)\ge 0$. in the \textit{full $k$-moment problem} for $\beta \equiv \beta^{(\infty)}$, a classical theorem of m. riesz ($n=1$) and e.k. haviland $(n>1$) shows that $\beta$ has a $k$-representing measure if and only if $l_{\beta}$ is $k$-positive. in the truncated $k$-moment problem, the direct analogue of riesz-haviland is not true. we discuss the gap between $k$-positivity and the existence of $k$-representing measures, with reference to tchakaloff's theorem, approximate $k$-representing measures, a ``truncated" riesz-haviland theorem due to curto-f., a ``strict" k-positivity existence theorem of f.-nie, and recent results concerning the \textit{core variety} of a multisequence.