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Isomorphic classification of symmetric spaces is an important problem relatedto the study of symmetric structures in arbitrary Banach spaces. This researchwas initiated in the seminal work of Johnson, Maurey, Schechtman and Tzafriri(JMST, 1979). Somewhat later it was extended by Kalton to lattice structures(1993). In particular, in JMST (see also Lindenstrauss-Tzafriri book [1979,Section 2.f]) it was shown that the space $L_2 \cap L_p$ for $2 \leq p <\infty$ (resp. $L_2+L_p$ for $1 < p \leq 2$) is isomorphic to $L_p$. A detailedinvestigation of various properties of separable sums and intersections of$L_p$-spaces (i.e., with $p<\infty$) was undertaken by Dilworth in the papersfrom 1988 and 1990. In contrast to that, we focus here on the problem if thenonseparable spaces $L_p +L_{\infty}$ and $L_p \cap L_{\infty}$, $1\lep<\infty$, are isomorphic or not. We prove that these spaces are not isomorphicif $1 \leq p < \infty$, $p \neq 2$. It comes as a consequence of the fact thatthe space $L_p \cap L_{\infty}$, $1\le p<\infty$, $p\ne 2$, does not contain acomplemented subspace isomorphic to $L_p$. In particular, as a subproduct, weshow that $L_p \cap L_{\infty}$ contains a complemented subspace isomorphic to$l_2$ if and only if $p = 2$. The problem if $L_2 +L_{\infty}$ and $L_2 \capL_{\infty}$ are isomorphic or not remains open.

$L_p+L_\infty$ and $L_p\cap L_\infty$ are not isomorphic for all $1\le p<\infty,$ $p\ne 2$